\section{Introduction}
\label{introduction}

Over the past few years the dominant explanation for the large mass to light ratios inferred for galactic and meta-galactic
systems, that these are embedded within massive dark matter halos, has begun to be challenged. Direct detection of the dark
matter particles, in spite of decades of extensive and dedicated searches, remains lacking. This has led some to interpret the
velocity dispersion measurements of stars in the local dSph galaxies (e.g. Angus 2008, Hernandez et al. 2010), the extended and
flat rotation curves of spiral galaxies (e.g. McGaugh, Sanders \& McGaugh 2002), the large velocity dispersions of galaxies
in clusters (e.g. Milgrom \& Sanders 2008), stellar dynamics in elliptical galaxies (e.g. Napolitano et al. 2012),
the gravitational lensing due to massive galaxies (e.g. Zhao \& Famaey 2010, Chiu et al. 2011, Mendoza et al. 2012),
and even the cosmologically inferred matter content for the universe through CMB and structure formation physics
(e.g. Skordis et al. 2006, Halle \& Zhao 2008, Milgrom 2012), not as indirect evidence for the existence of a dominant dark matter
component, but as direct evidence for the failure of the current Newtonian and General Relativistic theories of gravity, in the
large scale or low acceleration regimes relevant for the above situations.

Numerous alternative theories of gravity have recently appeared (e.g. TeVeS of Bekenstein 2004, and variations;
Bruneton \& Esposito-Farese 2007, Sobouti 2007, F(R) theories e.g. Capozziello \& De Laurentis 2011, Capozziello et al. 2007, conformal
gravity theories e.g. Mannheim \& Kazanas 1989), mostly grounded on geometrical extensions to General Relativity, and
leading to laws of gravity which in the large scale or low acceleration regime, mimic the MOdified Newtonian Dynamics (MOND)
fitting formulas. Similarly, Mendoza et al. (2011) have explored MOND not as a modification to Newton's second law, but as a
modified gravitational force law in the Newtonian regime, finding a good agreement with observed dynamics across galactic scales
without requiring dark matter. In fact, recently Bernal et al. (2011) have constructed an $f(R)$ extension to general relativity
which in the low velocity limit converges to the above approach.

A generic
feature of all of the modified gravity schemes mentioned above is the appearance of an acceleration scale, $a_{0}$, above which
classical gravity is recovered, and below which the dark matter mimicking regime appears. This last feature results in a general
prediction; all systems where $a>>a_{0}$ should appear as devoid of dark matter, and all
systems where $a<<a_{0}$ should appear as dark matter dominated, when interpreted under classical gravity.  It is interesting
that no $a>>a_{0}$ system has ever been detected where dark matter needs to be invoked, in accordance with the former condition.
On the other hand, the latter condition furnishes testable predictions. First, notice that for test
particles in orbit around a $1 M_{\odot}$ star, in circular orbits of radius $s$, the acceleration is expected to drop below
$a_{0}\approx 1.2 \times 10 ^{-10} m/s^{2}$ for $s>$7000 AU$=3.4\times10^{-2} pc$. The above provides a test for the dark matter/
modified theories of gravity debate; the relative velocities of components of binary stars with large physical separations should
deviate from Kepler's third law under the latter interpretation.

More specifically, seen as an equivalent Newtonian force law, beyond $s \approx $7000 AU the gravitational force
should gradually switch from the classical form of $F_{N}=GM/s^{2}$ to $F_{MG}=(G M a_{0})^{1/2}/s$, and hence the
orbital velocity, $V^{2}/s =F$, should no longer decrease with separation, but settle at a constant value, dependent
only on the total mass of the system through $V=(G M a_{0})^{1/4}$. That is, under modified gravity theories,
binary stars with physical separations beyond around 7000 AU should exhibit ``flat rotation curves'' and a
``Tully-Fisher relation'', as galactic systems in the same acceleration regime do.

In Hernandez et al. (2012a) we proposed that wide binary orbits may be used to test Newtonian gravity in the low acceleration regime.  There we applied
this test to the binaries of two very recent catalogues containing relative velocities and separations of wide binaries.
The two catalogues are entirely independent in their approaches.

At the somewhat larger scales of globular clusters, with sizes of tens of parsecs and masses of order $10^{5} M_{\odot}$,
the central values of the stellar velocity dispersion, projected on the plane of the sky, for many Galactic globular clusters
(GC) have been well known for decades, and are known to accurately correspond to the expectations of self-consistent dynamical models under
Newtonian gravity, e.g. King models (e.g. Binney \& Tremaine 1987, Harris 1996). Recently, a number of studies
(e.g. Scarpa et al. 2007a, 2007b, 2010 \& 2011 and Lane et al. 2009, 2010a, 2010b \& 2011, henceforth the Scarpa et al. and Lane et al. groups respectively)
have performed measurements of the projected velocity dispersion along the line of sight for stars in a number of Galactic GCs, but as a function of
radius, and reaching in many cases out to radial distances larger than the half-light radii of the clusters by factors of a few.

The surprising result of the above studies has been that radially, although velocity dispersion profiles first drop along Newtonian
expectations, after a certain radius, settle to a constant value, which varies from cluster to cluster.
As already noted by Scarpa et al. (2011), it is suggestive of a modified gravity scenario
that the point where the velocity dispersion profiles flatten, approximately corresponds to the point where average stellar accelerations
drop below $a_{0}$. Several recent studies have shown dynamical models for self-gravitating populations of stars under MOND or
other modified gravity variants (e.g. Moffat \& Toth 2008, Haghi et al. 2009, Sollima \& Nipoti 2010, Haghi et al. 2011, Hernandez \& Jim\'enez 2012)
which accurately reproduce not only the observed velocity dispersion profiles, but also the observed surface brightness profiles.

From the point of view of assuming Newtonian gravity to be exactly valid at all low velocity regimes, it has also been
shown that both velocity dispersion and surface brightness profiles for Galactic GCs can be self-consistently
modelled. Under this hypothesis, it is dynamical heating due to the overall Milky Way potential that is responsible for the
flattening of the velocity dispersion profiles e.g. Drukier et al. (2008), K\"{u}pper et al. (2010), Lane et al. (2010).
The constant velocity dispersion observed at large radii merely shows the contribution of unbound stars in the process of evaporating
into the Milky Way halo. In attempting to sort between these two contrasting scenarios, we took a fully empirical approach in Hernandez et al. (2012b).
There, we critically examine the plausibility of both gravitational scenarios by looking through the data for other correlations which each suggest.

For the Newtonian case, we examine the best available
inferences for the tidal radius of each cluster at closest galacto-centric passage, and compare it to the observed point
where the velocity dispersion flattens. We found the former to generally exceed the latter by factors of 4 on average,
making the Newtonian interpretation suspect. Also, we take all the clusters which the Lane et al. group have claimed
show no indication of a modified gravity phenomenology, based on the fact that their velocity dispersion profiles can be modelled
using Plummer profiles, and show that the fits with the generic asymptotically flat profiles we use are actually slightly better,
in all cases.

We shall use the term MONDian to refer to any modified theory of gravity which reproduces the
basic phenomenology of MOND in the low velocity limit for accelerations below $a_{0}$, of flat equilibrium velocities and a Tully-Fisher relation,
regardless of the details of the fundamental theory which might underlie this phenomenology.

Together with our
previous results of Hernandez \& Jimen\'ez (2012) showing that the asymptotic values of the velocity dispersion profiles
are consistent with scaling with the fourth root of the total masses, a Tully-Fisher relation for GCs, our results support
the interpretation of the observed phenomenology as evidence for a change in regime for gravity on crossing the $a_{0}$
threshold.

A recent study reaching the same conclusions, but at a significantly distinct scale, can be found in, Lee \& Komatsu (2010) show that
the infall velocity of the two components of the Bullet cluster, as required to
account for the hydrodynamical shock observed in the gas, is inconsistent with expectations of full cosmological simulations under
standard $\Lambda CDM$ assumptions. This has recently been confirmed at greater detail by Thompson \& Nagamine (2012), and can in fact be
seen as a failure not only of the $\Lambda CDM$ model, but of standard gravity, as the required collisional velocity is actually
larger than the escape velocity of the combined system. We note also the recent reviews by Kroupa et al. (2010), Famaey \& McGaugh (2012) and Kroupa (2012) and
references therein, detailing a number of observations in tension with standard $\Lambda CDM$ assumptions.

\section{Dynamics of wide binaries}

The Newtonian prediction for the relative velocities of the two components of binaries having circular
orbits, when plotted against the binary physical separation, $s$, is for a scaling of $\Delta V \propto s^{-1/2}$, essentially
following Kepler's third law, provided the range of masses involved were narrow.
In a relative proper motion sample however, only two components of the relative velocity appear, as velocity along the line of
sight to the binary leads to no proper motion. Thus, orbital projection plays a part, with systems having orbital planes along
the line of sight sometimes appearing as having no relative proper motions. A further effect comes from any degree of orbital
ellipticity present; it is hence clear that the trend for $\Delta V \propto s^{-1/2}$ described above, will only provide an upper
limit to the distribution of projected $\Delta V$ vs. $s$ expected in any real observed sample, even if only a narrow range of
masses is included. One should expect a range of measured values of projected $\Delta V$ at a fixed observed projected $s$, all
extending below the Newtonian limit, which for equal mass binaries in circular orbits gives $\Delta V_{N} =2 \left( \frac{G M}{s} \right)^{1/2}$.

Further, over time, the orbital parameters of binaries will evolve due to the effects of Galactic tidal forces and dynamical encounters
with other stars in the field, specially in the case of wide
binaries. To first order, one would expect little evolution for binaries tighter than the tidal limit of 1.7pc, and the
eventual dissolution of wider systems.
A very detailed study of all these points has recently appeared, Jiang \& Tremaine (2010). These authors numerically follow
populations of 50,000 $1 M_{\odot}$ binaries in the Galactic environment, accounting for the evolution of the orbital parameters
of each due to the cumulative effects of the Galactic tidal field at the Solar radius.  Also, the effects of close and long range
encounters with other stars in the field are carefully included, to yield a present day distribution of separations and relative
velocities for an extensive population of wide binaries, under Newtonian Gravity.

It is found that when many wide binaries cross their Jacobi radius, the two components remain fairly close by in both coordinate
and velocity space. Thus, in any real wide binary search
a number of wide pairs with separations larger than their Jacobi radii will appear. Finally, Jiang \& Tremaine (2010) obtain the
RMS one-dimensional relative velocity difference, $\Delta V_{1D}$, projected along a line of sight, for the entire
populations of binaries dynamically evolved over 10 Gyr to today, as plotted against
the projected separation on the sky for each pair. The expected Keplerian fall of $\Delta V_{1D} \propto s^{-1/2}$ for
separations below 1.7 pc is obtained, followed by a slight rise in $\Delta V_{1D}$ as wide systems cross the Jacobi radius
threshold. $\Delta V_{1D}$ then settles at RMS values of $ \approx 0.1 km/s$. This represents the best currently available estimate of how
relative velocities should scale with projected separations for binary stars (both bound and in the process of dissolving in
the Galactic tides) under Newtonian gravity.
We see that all we need is a large sample of relative proper motion and binary separation measurements to
test the Newtonian prediction for the RMS values of the 1 dimensional relative velocities of Jiang \& Tremaine (2010),
and to contrast the $\Delta V_{N} \propto s^{-1/2}$ and the $\Delta V_{MG} =cte.$ predictions for the upper envelope
of the $\Delta V$ vs. $s$ distributions.

\begin{figure}[!t]
\includegraphics[angle=0,scale=0.44]{Bin1.ps}
\caption{The figure shows projected relative velocities and separations for each pair of
wide binaries from the Shaya \& Olling (2011) {\it Hipparcos} catalogue having a probability of being
the result of chance alignment $<0.1$. The average value for the signal to noise ratio for the sample shown
is 1.7. The upper limit shows the flat trend expected from modified gravity theories, at odds
with Kepler's third law, shown by the $s^{-1/2}$ solid line.}
\end{figure}

\begin{figure}[!t]
\includegraphics[angle=0,scale=0.44]{Bin2.ps}
\caption{The figure shows projected relative velocities and separations for each pair of
wide binaries from the Dhital et al. (2010) SDSS catalogue within the distance range
($225<d/pc<338$). The average value for the signal to noise ratio for the sample shown
is 0.5. The upper limit shows the flat trend expected from modified gravity theories, at odds
with Kepler's third law, shown by the $s^{-1/2}$ solid line.}
\end{figure}

In the Shaya \& Olling (2011) catalogue wide binaries are identified by assigning a probability above chance
alignment to the systems by carefully comparing to the underlying background (and its variations) in a 5 dimensional
parameter space of proper motions and spatial positions. We keep only binaries with a probability
of non-chance alignment greater than $0.9$. The binary search criteria used by the authors requires that the proposed
binary should have no near neighbours; the projected separation between the two components is thus always many times smaller
than the typical interstellar separation. We use the reported distances to the primaries, where errors are smallest,
to calculate projected $\Delta V$ and projected $s$ from the measured $\Delta \mu$ and $\Delta \theta$ values reported.
Although the use of {\it Hipparcos} measurements guarantees the best available quality in
the data, we have also removed all binaries for which the final signal to noise ratio in
the relative velocities was lower than $0.3$.

We are left with a sample of 280 binaries, having distances to the Sun within
$6<d<100$ in pc. The data show a perfectly flat upper envelope in a $\Delta V$ vs. projected $s$, Hernandez \& Jim\'enez (2012).
The average signal to noise ratio for the data is 1.7, with an average error on $\Delta V$ of 0.83 $km/s$,
which considering a $2 \sigma$ factor from the top of the distribution to the real underlying upper limit for the sample,
results in 3 $km/s$ as our estimate of the actual physical upper limit in $\Delta V$.

The Sloan low mass wide pairs catalogue (SLoWPoKES) of Dhital et al. (2010) contains a little over 1,200 wide
binaries with relative proper motions for each pair, distances and angular separations. Also, extreme care was
taken to include only physical binaries, with a full galactic population model used to exclude chance alignment
stars using galactic coordinates and galactic velocities, resulting in an estimate of fewer than 2\% of false
positives. This yields only isolated binaries with no neighbours within many times the internal binary separation.
Again, we use the reported distances to the primaries to calculate
projected $\Delta V$ and projected $s$ from the measured $\Delta \mu$, $\Delta \theta$ and $d$ values reported by Dhital et al. (2010),
to obtain a sample of 417 binaries.

The upper envelope of the distribution of
$\Delta V$ from this catalogue does not comply with Kepler's third law. As was the case with the {\it Hipparcos}
sample, the upper envelope describes a flat line, as expected under modified gravity
schemes. The average signal to noise in $\Delta V$ for the Dhital et al. (2010) catalogue is
0.48, with an average error on $\Delta V$ of 12 $km/s$, which considering a $2 \sigma$ factor from the top
of the complete distribution to the real underlying upper limit gives the same $3 km/s$ as obtained for the Shaya \& Olling (2011)
{\it Hipparcos} catalogue.

\begin{figure}[!t]
\includegraphics[angle=0,scale=0.44]{Bin4.ps}
\caption{The solid curve gives the RMS values for one dimensional projected relative velocities as a function of
projected separations, for the detailed dynamical modelling of large populations of wide binaries evolving in
the Galactic environment, taken from Jiang \& Tremaine (2010). The same quantity for the data from the catalogues analysed is
given by the points with error bars; those with narrow $log(s)$ intervals being from the {\it Hipparcos} sample of Shaya \& Olling (2011),
and those two with wide $log(s)$ intervals coming from the SDSS sample of Dhital et al. (2010).}
\end{figure}

Figure (3) shows the RMS value of the one-dimensional relative velocity difference for
both of the samples discussed. The error bars give the error propagation on $\Delta \mu$ and $d$. We construct $\Delta V_{1D}$
by considering only one coordinate of the two available from the relative motion on the
plane of the sky. Thus, each binary can furnish two $\Delta V_{1D}$ measurements, which statistically should not introduce any
bias. Indeed, using only $\Delta \mu_{l}$ or only $\Delta \mu_{b}$ or both for each binary, yields the same mean values for the
points shown. The small solid error bars result from considering an enlarged sample where each binary contributes two $\Delta V_
{1D}$ measurements, while the larger dotted ones come from considering each binary only once, and do not change if we consider
only $\Delta \mu_{l}$ or only $\Delta \mu_{b}$. The series of small $log(s)$ interval data are for the {\it Hipparcos} catalogue
of SHaya \& Olling (2011), while the two broader crosses show results for the Dhital et al. (2010) SDSS sample.

The solid curve is the Newtonian prediction of the full Galactic evolutionary model of Jiang \& Tremaine (2010) for
binaries, both bound and in the process of dissolving. Note that the results of this simulation deviate from Kepler's law
for $s$ larger than the Newtonian Jacobi radius of $1.7 pc$, whereas the discrepancy with
the observed samples also occurs at much smaller separations.  Even considering the large error bars, where
each binary contributes only one $\Delta V_{1D}$ value, we see eight points lying beyond 1$\sigma$, making the probability of
consistency between this prediction and the observations of less than $(0.272)^{8}$=$3\times 10^{-5}$.

We obtain a constant RMS value for $\Delta V_{1D}$ of 1 $km/s$, in qualitative agreement with expectations
from modified gravity schemes. The vertical line marks $a=a_{0}$; we see the data departing from the
Newtonian prediction outwards of this line, and not before.
The two independent catalogues, each using different sets of selection criteria, each
perhaps subject to its own independent systematics, are consistent with the same result, a constant horizontal upper envelope for
the distribution of relative velocities on the plane of the sky at an intrinsic value of 3 $km/s \pm$1 $km/s$, extending over 3
orders of magnitude in $s$, with a constant RMS $\Delta V_{1D}$ value consistent with 1 $km/s \pm$0.5 $km/s$. This supports the
interpretation of the effect detected as the generic prediction of modified gravity theories.

\section{Outer dynamics of globular clusters}

We begin by modelling the observed projected radial velocity dispersion profiles, $\sigma_{obs}(R)$, for the globular clusters in our sample,
Hernandez et al. (2012b). As seen from the Scarpa et al. and Lane et al. data, the observed velocity dispersion profiles show a central core
region where the velocity dispersion drops only slightly, followed by a ``Keplerian'' zone where the drop is more pronounced.
These first two regions are in accordance with standard Newtonian King profiles, but they are then followed by a third outermost
region where the velocity dispersion profiles cease to fall along Keplerian expectations, and settle to fixed values out to the last
measured point. As some of us showed in Hernandez \& Jim\'enez (2012), an accurate empirical modelling for these velocity dispersion
profiles can be achieved through the function:

\begin{figure}[!t]
\includegraphics[angle=0,scale=0.44]{GC1.ps}
\caption{
The figure shows the observed projected velocity dispersion profiles for six representative GCs in our sample,
points with error bars, as a function of the radial coordinate, normalised to the half-light radius of each. The solid curves
give the maximum likelihood fits to the asymptotically flat $\sigma(R)$ model of eq.(1), seen to be accurate descriptions of the data.
The vertical lines indicate the $a=a_{0}$ threshold, and the arrows the point where the profiles flatten, {\it a priori}
independent features, in most cases seen to occur at approximately the same place.}
\end{figure}

\begin{equation}
\sigma(R)= \sigma_{1} e^{-(R/R_{\sigma})^{2}} + \sigma_{\infty}
\end{equation}

In the above equation $\sigma_{\infty}$ is the asymptotic value of $\sigma(R)$ at large radii, $R_{\sigma}$ a scale radius fixing how
fast the asymptotic value is approached, and $\sigma_{1}$ a normalisation constant giving $\sigma(R=0)=\sigma_{1}+\sigma_{\infty}$.

We now take the observed data points $\sigma_{obs}(R_{i})$ along with the errors associated to each data point, to determine
objectively through a maximum likelihood method the best fit values for each of the three parameters in equation (1), for each
of the 16 observed globular clusters. The confidence intervals for each of the three parameters are then
obtained without imposing any marginalisation. This last point allows to properly account for
any correlations between the three fitted parameters when calculating any quantity derived from combinations of them, as will
be constructed in what follows.

Taking $\sigma_{obs}(R_{i})$ data from Drukier et al. (1998), Scarpa et al. (2004), (2007a), (2007b), (2010) and (2011),
Lane et al. (2009), (2010a), (2010b) and (2011) and half-light radii, $R_{1/2}$, from integrating the surface density brightness profiles of Trager et al.
(1995), we perform a maximum likelihood fit for all the sixteen globular clusters studied.

Figure (4) shows the observed projected velocity dispersion profiles for 6 representative
globular clusters from our sample, points
with error bars. The radial coordinate has been normalised to the $R_{1/2}$ radius of each of the clusters.
The continuous curves show the maximum likelihood fits for each cluster, which are clearly good representations
of the data. We can now give $R_{f}=1.5R_{\sigma}$ as an adequate empirical estimate of the radius beyond which the
dispersion velocity profile becomes essentially flat. In terms of equation (1), which can be seen to be highly consistent
with the observed velocity dispersion profiles, $R_{f}$ is the radius such that $\sigma(R_{f})=0.1\sigma_{1}+\sigma_{\infty}$,
a good representation of the transition to the flat behaviour, as can be checked from figure (4), where the arrows
give $R_{f}$, with the horizontal lines on the arrows showing the $1\sigma$ confidence intervals on these fitted parameters.
An empirical definition of the radius where the typical acceleration felt by stars drops below $a_{0}$ can now
be given as $R_{a}$, where:

\begin{equation}
\frac{3 \sigma(R_{a})^{2}}{R_{a}}=a_{0}.
\end{equation}

Using the above definition, we can now identify $R_{a}$ for each of the globular clusters studied. The vertical lines in Figure(5)
show $R_{a}$ for each cluster, also normalised to the half-light radius of each. In the figure, clusters have been ordered by their
$R_{a}/R_{1/2}$ values, with the smallest appearing at the top, and $R_{a}/R_{1/2}$ growing towards the bottom of the figure.

The good fits shown in the studies of the Lane et al. group to the observed velocity dispersion profiles using
Plummer models are clearly not sufficient to dismiss a modified gravity interpretation, as the asymptotically flat projected dispersion
velocity fits of the type used for full dynamical modelling under modified gravity (Hernandez \& Jimenez 2012) actually provide even slightly
better fits to the data, Hernandez et al. (2012b).

\begin{figure}[!t]
\includegraphics[angle=0,scale=0.44]{GC2.eps}
\caption{
The figure shows the relation between the point where the velocity dispersion flattens, $R_{f}$, and the Newtonian tidal radius,
$R_{T}$, for each cluster. Even considering the large errors involved on both quantities, most points fall far to the right of the
identity line shown, making the Newtonian explanation for the flattened velocity dispersion profiles, rather suspect.}
\end{figure}

It is interesting at this point to notice a first correlation, the smaller the value of $R_{a}/R_{1/2}$, the larger the
fraction of the cluster which lies in the $a<a_{0}$ regime, and interestingly, the flatter the velocity dispersion
profile appears. At the top of the figure we see clusters where stars experience accelerations below $a_{0}$ almost at all radii,
and it so happens, that it is only in these systems that the velocity dispersion profile appears almost flat throughout.
Towards the bottom, we see systems where only at the outskirts accelerations fall under $a_{0}$. Over most of their extents,
these clusters lie in the Newtonian $a>a_{0}$ regime, and indeed, it is exclusively these, that show a clear Keplerian
decline in the projected velocity dispersion profiles over most of their extents. Also, notice that $R_{f}$ and $R_{a}$
approximately coincide, as already previously noticed by Scarpa et al. (2007a), the flattening in the velocity dispersion
profiles seems to appear on crossing the $a_{0}$ threshold.

In order to test the validity of the explanation for the outer flattening of the observed velocity dispersion profiles under
Newtonian gravity, that these indicate dynamical heating due to the tides of the Milky Way system (bulge plus disk plus dark halo),
we need accurate estimates of the Newtonian tidal radii for the clusters studied.  One of us in Allen et al. (2006) and
Allen et al. (2008) performed detailed orbital studies for 54 globular clusters for which absolute proper motions and line of sight
velocities exist. In that study, both a full 3D axisymmetric Newtonian mass model for the Milky Way and a model incorporating a galactic bar
were used to compute precise orbits for a large sample of globular clusters, which fortunately includes
the 16 of our current study. The Galactic mass models used in those papers are fully consistent with all kinematic and structural
restrictions available. Having a full mass model, together with orbits for each globular cluster, allows the calculation of the
Newtonian tidal radius, not under any ``effective mass'' approximation, but directly through the calculation of the derivative
of the total Galactic gravitational force, including also the evaluation of gradients in the acceleration across the extent of the
clusters, at each point along the orbit of each studied cluster.

The Newtonian tidal radii we take for our clusters, $R_{T}$, are actually the values which results in the largest dynamical heating
effect upon the clusters studied, those at perigalacticon. As the distance of closest approach to the centre of the Galaxy might
vary from passage to passage, as indeed it often does, detailed orbital integration is used to take $R_{T}$ as an average
for perigalactic passages over the last 1 Gyr.

\begin{figure}[!t]
\includegraphics[angle=0,scale=0.44]{GC3.eps}
\caption{Here we give the relation between the observed asymptotic dispersion velocity measurements, and the total mass of each
cluster. The line gives the best fit $\sigma \propto M^{1/4}$ scaling for the data.}
\end{figure}

We update the tidal radii published in Allen et al. (2006) and Allen et al. (2008),
by considering revised total masses from the integration of the observed $V$ band surface brightness profiles for our clusters
(Trager et al. 1995), and using the $V$ band stellar $M/L$ values given in McLaughlin \& van der Marel (2005) and accompanying electronic tables.
For each individual GC, detailed single stellar population models tuned to the inferred ages and metallicities of each of the
clusters we model were constructed in that study, using various standard population synthesis codes, and for a variety of assumed
IMFs. In this way present stellar $M/L$ values in the $V$ band were derived, which we used. As we did not in any way use the dynamical
mass estimates of McLaughlin \& van der Marel (2005), the total masses we used are independent
of any dynamical modelling or assumption regarding the law of gravity, as they are derived through completely independent surface brightness profile
measurements and stellar population modelling. The confidence intervals in our tidal
radii include the full range of stellar $M/L$ values given by McLaughlin \& van der Marel (2005), through considering a range
of ages, metallicities and initial mass functions consistent with the observed HR diagrams for each cluster.

In Figure (5) we show values of $R_{f}$ for our clusters, plotted against their corresponding $R_{T}$ values, both in units of
pc. The error bars in $R_{f}$ come from full likelihood analysis in the fitting process of equation (1) to
$\sigma_{obs}(R)$, which guarantees that confidence intervals in both of the quantities plotted are robust $1\sigma$ ranges.
The solid line shows a $R_{f}=R_{T}$ relation. It is obvious from the figure that the onset of the flat velocity dispersion regime
occurs at radii substantially smaller than the tidal radii, for all of the globular clusters in our sample. Even under the most
extreme accounting of the resulting errors, only three of the clusters studied are consistent with $R_{T} \approx R_{f}$ at $1\sigma$. Actually,
the average values are closer to  $R_{T}=4 R_{f}$, with values higher than 8 appearing. One of the clusters, NGC 5024 does not appear, as
it has values of $R_{T}=184.12$, $R_{f}=36$, which put it out of the plotted range, but consistent with the description given above.
Given the $R^{3}$ scaling of Newtonian tidal phenomena, even a small factor of less than 2 inwards of the tidal radii, tides can be
safely ignored, e.g. in Roche lobe overflow dynamics, the stellar interior is largely unaffected by the tidal fields, until almost
reaching the tidal radius. It therefore appears highly unlikely under a Newtonian scheme, that Galactic tides could be responsible
for any appreciable dynamical heating of the velocity dispersion of the studied clusters.

We note that Lane et al. (2010) and Lane et al. (2012) find that Newtonian tidal heating can explain the observed velocity dispersion
profile of their GC sample. However, it is important to note that in Lane et al. (2010)  and Lane et al. (2012), total masses were calculated
directly from the observed velocity dispersion observations, under the assumption that Newtonian dynamics hold. If that assumption is to be tested,
the importance of deriving total masses through an independent method, not based on stellar dynamics, is evident.

Notice also, that most of the clusters in our sample are problematic for
a Newtonian gravity scheme, even without the recent observations of an outer flat velocity dispersion profile. As remarked
already in Allen et al. (2006), the clusters in our sample have Newtonian tidal radii larger than the observed truncation radii
of their light distribution, the sole exceptions being Omega Cen (NGC 5139) and M92 (NGC 6341), two rather anomalous clusters.
Whereas a full dynamical modelling under an extended gravity force law of these clusters, Hernandez \& Jim\'enez (2012), naturally
yielded an outer truncation for the light profile, under a Newtonian hypothesis, the observed truncation in the light profile of
the clusters in our sample cannot be explained as arising from interaction with the tidal field of the Milky Way.

Furthermore, notice that we have taken $R_{T}$ at perigalacticon, where tides are at their most severe over the clusters orbit, any
other orbital occupation averaging would result in substantially larger $R_{T}$ values. Notice also that as shown by Allen et al. (2006)
and Allen et al. (2008), the inclusion of a realistic massive Galactic bar potential, in the case of the clusters in our sample, results
generally in negligible changes in the resulting $R_{T}$ values, or in some cases, a slight increase in these values. Hence, even taking
the fullest non-axisymmetric Galactic mass model under Newtonian gravity, with precise orbits derived from 3D velocity measurements for
the clusters studied, together with total mass determinations tuned to the individual stellar populations of them, yields tidal radii as
shown in figure (5).

As already noticed by Scarpa et al. (2011), the flattening
in the observed velocity dispersion profiles seems to appear at the point where the $a_{0}$ threshold is crossed. Here we use the much more
careful and objective modeling of the observed velocity dispersion curves of the previous section to test this point.

We see that most GCs in the sample fall within $1\sigma$ of the identity, Hernandez et al. (2012b), with about a third falling further away.
A quantitative test of the correlation being explored is possible, since the careful modelling of the velocity dispersion profiles
we performed naturally yields objective confidence intervals for the parameters of the fit.

We end this section with Figure (6), which shows the relation between the measured asymptotic velocity dispersion, $\sigma_{\infty}$,
and the total mass of the clusters in question. These masses represent
the best current estimates of the stellar mass for each of the clusters in the sample, including corresponding confidence intervals.
As with all the other correlations and data presented, there is no dynamical modelling or modified gravity
assumptions going into Figure (6), merely observable quantities. We see, as already pointed out
in Hernandez \& Jim\'enez (2012), the GCs observed nicely comply with a scaling of $\sigma \propto M^{1/4}$, the Tully-Fisher
law of galactic systems ``embedded within massive dark haloes''.

The straight line shows the best fit $\sigma \propto M^{1/4}$ scaling, and actually falls only a factor of 1.3 below the modified gravity
prediction for systems lying fully within the low acceleration regime (e.g. Hernandez \& Jim\'enez 2012), for the same value of
$a_{0}=1.2 \times 10^{-10} m/s$ used here, as calibrated through the rotation curves of galactic systems. This small offset is not surprising,
since the GCs treated here are not fully within the $a<a_{0}$ condition, most have an inner Newtonian region encompassing a substantial
fraction of their masses. A preliminary version of this last figure appeared already in Hernandez \& Jim\'enez (2012); we reproduce here an
updated version using now the extended sample of clusters treated, and $\sigma_{\infty}$ values and their confidence intervals as derived through
the careful velocity dispersion fitting procedure introduced.

To summarise, we have tested the Newtonian explanation of Galactic tides as responsible for the observed $\sigma(R)$ phenomenology,
and found it to be in tension with the observations, given the tidal radii (at perigalacticon) which the GCs in our sample present, are generally
larger than the points where  $\sigma(R)$ flattens, on average, by factors of 4, with values higher than 8 also appearing. An explanation under a
MONDian gravity scheme appears likely, given the clear correlations we found for the clusters in our sample, all in the expected sense.

\section{The bullet cluster and other inconcistencies of GR at galactic scales}

In going to galactic scales, the local dwarf spheroidal satellites (dSph) of the Milky Way offer an interesting test bed for
modified theories of gravity, being characterised by the highest dark matter fractions, when modelled under standard
gravity. In Hernandez et al. (2010) we showed that fully self consistent dynamical models can be constructed for these objects under
a modified Newtonian force law, and found an interesting correlation. Assuming standard gravity, dSphs
with the oldest stellar populations show the highest dark matter fractions, something which has to be regarded as a curious coincidence.
However, under the assumption that the stars alone determine the gravitational potential of modified gravity schemes, it is natural to
expect, indeed it is predicted, that the gravitational forces will be strongest, and deviate further from the Newtonian predictions,
for the oldest stellar populations, having higher intrinsic mass to light ratios.

More recently, Kroupa (2012) has pointed out a serious inconsistency in the standard gravity interpretation of tidal dwarf galaxies.
These small systems are formed during the interaction and merger processes of large galaxies. Within standard gravity, they are seen as
transient stellar structures displaying out of equilibrium dynamics and containing no dark matter. This last condition follows from
the large velocity dispersion values of the dark matter in the haloes of the large colliding galaxies, more than an order of
magnitude larger than what would be required to form a stable halo around the forming small tidal dwarf galaxies. However, recent
observations have confirmed that these tidal dwarf galaxies have the same dynamics as normal, stable dwarf galaxies showing
relaxed, equilibrium dynamics ``dominated by dark matter halos''. Basically, tidal dwarf galaxies lie along the same Tully-Fisher
relation as defined by normal galaxies, including more standard dwarfs. Thus, under a standard gravity plus dark matter interpretation,
we must accept as a curious coincidence that tidal dwarf galaxies, transient swirls of tidally drawn material in interacting galaxies,
have the same internal velocities as normal dwarf galaxies, equilibrium systems dominated by dark matter haloes. Under a MONDian
modified view however, the observation above is a prediction; once tides form low density ensembles of stars, these will behave
just like any other such set of stars, a standard dwarf galaxy dominated by the modified gravity regime, as internal accelerations
are below $a_{0}$.

At the even largest Mpc scales of the bullet cluster, a serious inconsistency in the standard gravity interpretation has been pointed out
by Lee \& Komatsu (2010). These authors show that the encounter velocities for the two components of the bullet cluster, as determined
by the careful hydrodynamical modelling of Mastropietro \& Burkert (2008) constrained to yield the observed bow shock in the glowing x-ray gas, are so high, that they
are totally incompatible with the current standard cosmological scenario. This conclusion has recently been confirmed in much more detail by
Thompson \& Nagamine (2012), who show clearly through full standard cosmological simulations, that the probability of finding
such a system as the observed bullet cluster, at the observed redshift, is lower than $5 \times 10^{-8}$, essentially a statement
of the impossibility of producing such a system under the standard cosmological scenario. The origin of these inconsistency is clear,
and stems from the relation between the sound speed for the gas inside a virialized halo, and the escape velocity of these same halo.
This last velocity is the highest at which an object falling into the central more massive cluster can be expected to appear, and it is
only a factor of 2 higher than the isothermal sound speed in the gas. Thus, encounters with Mach numbers higher than two are inconsistent
with standard gravity. Within an expanding cosmological scenario, one must first overcome the expansion, making the highest Mach numbers
possible, somewhat lower than two. The strong bow shock seen in the x-ray gas of the bullet cluster however, testifies to an encounter
with a Mach number higher than 2, and hence inconsistent with standard GR and Newtonian gravity. Clearly, no amount of dark matter
can alleviate the problem, as adding dark matter will increase the impact velocity, but increase also, in the same proportion, the
sound speed in the gas before the collision, and hence leave the Mach number of the shock unchanged. One can not simply say that both
original clusters were orbiting within the higher gravitational potential of a much larger structure, as at that redshift of 0.5, in general,
no larger structures excised, indeed, none is seen in the vicinity of the cluster in question.

\section{Conclusions}

We have reviewed recent dynamical observations for wide binaries in the solar neighbourhood which show clear departures
from the Newtonian predictions on crossing the $a_{0}$ threshold, with equilibrium velocities becoming constant, as
expected under MONDian modified gravity schemes calibrated to explain rotation curves of large galaxies without the
need for any dark matter. Under a standard gravity interpretation, adding dark matter is not an option, as the scales
involved are smaller than any where dark matter is expected, and any such addition would have to be extremely fine tuned;
no dark matter for binaries having accelerations higher than $a_{0}$, and an increasing mini halo necessary
as the semi-major axis grows beyond 7000 AU.

We have shown that the velocity dispersion profiles of globular clusters in the Milky Way tend to constant velocity
values at the outskirts, at values which happen to scale precisely along the same $M^{1/4}$ relation as the galactic
Tully-Fisher relation. This is natural under any MONDian gravity scheme, but is inconsistent with standard gravity,
where tidal velocity disruptions show an inverse scaling with mass, and clear scalings with orbital parameters and
internal concentrations, none of which appear in the globular clusters studied.

At the largest scales of the colliding galaxy clusters in the bullet cluster system, it has been shown that
the standard cosmological scenario is incapable of producing the strong bow shock seen in the glowing
x-ray gas. Adding any choice amount of dark matter will not solve the problem, as the increase in the infall
velocity will be accompanied by an increase in the pre-collision sound speed of the gas, cancelling any potential
increase in the Mach number of the collision, necessary to explain the observed shock.

The recent body of observational evidence reviewed her forces a change from considering GR plus dark matter, or modified gravity,
as equivalently plausible options; in the low velocity, $a<a_{0}$ regime, gravity does not follow the standard descriptions of
Einstein and Newton.

\section{Acknowledgements}

Xavier Hernandez acknowledges financial assistance from UNAM DGAPA grant IN103011. Alejandra Jim\'enez
acknowledges financial support from a CONACYT scholarship.


\section{Introduction}

It is well known that ferromagnetic (FM) systems of discrete spins with a finite range of interaction have a lower critical dimension $D_L=1$.
It means that a one-dimensional chain of spins can not undergo a phase transition at any positive temperature \cite{D1SR}.
The situation is different if one considers long-range (LR) models \cite{Dyson} in $d$ dimensions.
They are fully connected models, with a Hamiltonian:
\begin{equation}\label{Eq:H}
 H=-\frac{1}{2}\sum_{i,j=1}^{N}J_{ij}\sigma_i\sigma_j\;.
\end{equation}
The range of interactions is infinite and the intensity of the coupling $J_{ij}$
decays as a power law with the distance between spins: $J_{ij}\propto|r_{ij}|^{-(d+\sigma)}$.
One can also define spin glasses on LR models, taking $J_{ij}$ as independent identically distributed random variables,
extracted from a distribution $P(J)$ (like for example a binary or Gaussian distribution),
requiring that the variance of $P(J)$ decays as a power law:
$\overline{J_{ij}^2}\propto |r_{ij}|^{-(d+\sigma)}$ \cite{LR-SG}.
These models can have a transition at dimensions smaller than the lower critical one for usual short range (SR) models.
Indeed a ferromagnetic LR model can have a transition also in $d=1$. Furthermore, varying $\sigma$,
the behaviour of the system (such as for example the critical exponents) can vary from a mean field to a non mean field one
until it reaches a certain value $\sigma_L$ and the behaviour of the corresponding SR system is recovered.
In fact one can write down a relation between $(d,\sigma)$ and the effective dimension $D$ of an equivalent SR model.
At $\sigma_L$, the effective dimension $D$ reduces to the real dimension $d$ of the LR system.
The behaviour in and out the range of validity of mean field approximation can thus be observed varying only a parameter,
and this is very useful if one wants to simulate the system numerically because the computational complexity of the model does not change
with the effective dimension.
Different LR models have been introduced in the past, and more than one relation $(d,\sigma) \leftrightarrow D$  exist.
Nonetheless, are still unclear the differences between various LR models,
the exactness of the $(d,\sigma) \leftrightarrow D$  relations and their limits.
Often LR models (both ferromagnetic and disordered ones)
have been used to extract properties of the analogous SR effective models, however it is not clear whether
this operation is really justified.

The purpose of this work is to summarize the previous works on LR models,
and to answer some crucial questions on how good are LR models to simulate SR models,
which is the best $(d,\sigma) \leftrightarrow D$  relation, what is its range of validity and how similar are different LR models.
Many of the answers are unknown even in the simplest ferromangetic case, for this reason we will mostly analyze FM systems.

\section{Review of known analytical and numerical results for ferromagnetic Long Range models}

The simplest LR model that can undergo a paramagnetic/ferromagnetic phase transition is a one-dimensional
chain of spins with the Hamiltonian of Eq.~(\ref{Eq:H}) and $J_{ij}\propto |i-j|^{-(1+\sigma)}$.
For this model, Dyson demonstrated analytically that there is a standard second order phase transition if $0<\sigma<1$ \cite{Dyson}.
For $\sigma\leq 0$ the energy is no longer an extensive quantity.

This model can be easily generalized to $d$ dimensions, redefining the couplings as $J_{ij}\propto r_{ij}^{-(d+\sigma)}$ where $r_{ij}$ is the euclidean distance $r_{ij}=|\vec{r}_i-\vec{r}_j|$.

This model has been analyzed using a renormalization group (RG) approach in Ref.~\cite{fieldtheoryLR}.
The field theory in the momentum space can be written as:

\begin{equation}
\begin{aligned}
 \int dx \mathcal{L}(\phi)= \sum_ku_2(k)\phi(k)\phi(-k)+u\sum_{k_1 k_2 k_3}\phi(k_1)\phi(k_2)\phi(k_3)\phi(-k_1-k_2-k_3)
\label{Eq:LRLagrangian}
\end{aligned}
\end{equation}
where $u_2(k)=r+j_{\sigma}k^{\sigma}+j_2k^2$, and the parameter $r$ varies linearly with the temperature, being null at criticality.

The intuitive, but too naive, interpretation given in Ref.~\cite{fieldtheoryLR} is the following:
if $\sigma>2$, the leading term in $u_2(k)$ is the $k^2$ one, then the usual SR behaviour in $d$ dimensions is recovered; while, for $\sigma<2$, the leading term in $u_2(k)$ is the $k^{\sigma}$ one, the $k^2$ term being subleading, and a behaviour different from the SR one is present.

If $2\sigma-d<0$ the Gaussian fixed point $u^*=r^*=0$ is stable.
The critical exponents are easily calculated, leading to $\nu=1/\sigma$, $\eta=2-\sigma$, $\gamma=1$.
At $\epsilon=2\sigma-d=0$, this fixed point is marginally stable and logarithmic behaviour appears for the correlation length and susceptibility.
This point corresponds to the upper critical exponent $\sigma_U=\frac{d}{2}$.
For $\epsilon>0$ the Gaussian fixed point is unstable with respect to $u$ and a new fixed point $u^*=O(\epsilon)$ is found.
The critical exponents can be obtained as a series expansion in $\epsilon$.
$\eta$ is found to be not renormalized up to third order in $\epsilon$ and it is commonly believed that
it will have the mean-field value at all orders because new $k^{\sigma}\phi(k)\phi(-k)$ terms are not generated under renormalization.
This has been also verified numerically with good accuracy in Ref.~\cite{LR_SRnumeric}.

Summarizing, the picture that emerges from the work of Ref.~\cite{fieldtheoryLR} is the following:
for $0<\sigma<\sigma_U=\frac{d}{2}$ the system is in a mean-field region, for $\sigma_U<\sigma<\sigma_L=2$
the exponents are different from the mean field ones and
change continuously with $\sigma$, for $\sigma>\sigma_L$ the SR behaviour is recovered.
However there are some debated points.
For example, in this picture the lower critical exponent is $\sigma_L=2$, for all the dimensions\footnote{\baselineskip 15pt
In SR systems the lower critical dimension is the dimension at which the phase transition ceases to exist.
In LR models, we call lower critical exponent the value of $\sigma$ such that the SR behaviour is recovered.
If the SR model has a phase transition in $d$ dimensions, then there is a transition also for $\sigma>\sigma_L$; on the contrary, as in $d=1$, there is no transition for $\sigma>\sigma_L$.}.
However in $d=1$, at $\sigma=1$ the transition becomes of the Kosterlitz-Thouless (KT) type \cite{KT},
supported by analytical \cite{LCD-analytical} \cite{LCD-analytical2} and numerical \cite{LCD-numerical} evidences.

The problem of the inconsistency of the results of \cite{fieldtheoryLR} near $\sigma=2$
is not related only to the one-dimensional case.
In fact, according to the picture of Ref.~\cite{fieldtheoryLR}, $\eta=2-\sigma$ for $\sigma<2$ and $\eta=\eta_{SR}$, for $\sigma>2$.
This would imply a jump discontinuity in $\eta$ at $\sigma = 2$, and a non-monotonic behaviour in $\sigma$. While this
phenomenon is not forbidden by thermodynamic arguments (which only require $\eta\leq2+\sigma$), it has
attracted considerable attention over the past decades, because it is quite singular.

\begin{figure}[t]
\begin{center}
\includegraphics[width=0.7\columnwidth]{eta}
\end{center}
\caption{\label{Fig:eta} Behaviour of $\eta(\sigma)$ for a LR model in $d=2$ as proposed in different works: in the work of Fisher et al.~\cite{fieldtheoryLR}	 $\eta=2-\sigma$ up to $\sigma=2$, while for Sak \cite{Sak} $\eta=\max(2-\sigma,\eta_{SR}=\frac{1}{4})$, and the data of Picco \cite{Picco} $\eta$ seem to interpolate smoothly between $2-\sigma$ and $\eta_{SR}=\frac{1}{4}$.}
\end{figure}

In Ref.~\cite{Sak} a different scenario was proposed. In fact, if the term $j_2k^2$ is not ignored in Eq. (\ref{Eq:LRLagrangian}) when $\sigma<2$,
as done in Ref.~\cite{fieldtheoryLR}, it can be seen that the non-trivial fixed point is characterized by $j_2^*=O(\epsilon^2)\neq0$.
Even if one starts with $j_2=0$, SR forces appear after the renormalization, determining the critical behaviour.
As a consequence, for $d < 4$ the boundary between the intermediate and the SR regime was found to shift from
$ \sigma_L= 2$ to $\sigma_L = 2-\eta_{SR}$.
In particular, for $\sigma<2-\eta_{SR}$, the introduction of $j_2\neq0$ does not change the critical exponents, that remain those of
Ref.~\cite{fieldtheoryLR}. When $\sigma>2-\eta_{SR}$, all the exponents become the SR ones, without discontinuity, and without loosing
the monotonicity in $\sigma$.
In fact in this regime the fixed point is characterized by $j_{\sigma}^*=0$, and the field theory is the usual SR one.

In support to this picture, in a field-theoretic approach, Honkonen and Nalimov \cite{Honkonen} proved,
to all orders in perturbation theory, the stability of the SR fixed point for $\sigma> 2-\eta_{SR}$
and of the LR one for $\sigma< 2-\eta_{SR}$.
Within this new scenario, the theory is also consistent with the exact results for the one-dimensional case.
In fact, for $d=1$, $\eta_{SR}=1$. In this way the lower critical exponent is $\sigma_L=1$, as expected.
However, the analysis of \cite{Sak} has also been the subject of
criticism: in Ref.~\cite{vanEnter} the results for $n\geq2$ are contested,
in Ref.~\cite{Yamazaki} the absence of the kink at $\sigma=2-\eta_{SR}$ is hypothesized, in Ref.~\cite{Gusmao}
the picture of Ref.~\cite{fieldtheoryLR} is supported.
All this works on the subject are related to the importance of understanding how to treat systems in presence of different,
competing fixed points.

There are also numerical studies.
In Ref.~\cite{LR_SRnumeric} a Monte Carlo study of a LR model in $d=2$, using cluster algorithms, supports the scenario of Ref.~\cite{Sak} where
$\eta=\max(2-\sigma,\eta_{SR}=\frac{1}{4})$,
excluding definitively the picture of Ref.~\cite{fieldtheoryLR}.
In particular they affirm to find logarithmic corrections to scaling at $\sigma=1.75$,
clear indication of a crossover between different critical points.
Very recently, in Ref.~\cite{Picco} the same study has been improved.
In fact, the measurement of $\eta$ for a LR system in $d=2$ has been repeated,
close to the region where its behavior is changing, i.e. for $\sigma\simeq2 -\eta_{SR}$, obtaining more precise results.
The author of Ref.~\cite{Picco} confirms that there is no
discontinuity but a clear deviation from the behavior predicted by Sak in Ref.~\cite{Sak} is measured.
In particular in the intermediate regime up to $\sigma \simeq 1.5$ the results are
in agreement with the prediction $\eta=\eta_{LR}=2-\sigma$. For $\sigma>2$, $\eta$ is in perfect agreement with the
value for a SR model. In the remaining part for $1.6 \leq\sigma\leq 2$ the results do not agree
with the prediction of the RG analysis \cite{fieldtheoryLR}\cite{Sak}. On the
contrary, $\eta$ seems to interpolate smoothly between these two behaviors.
This behaviour is also supported by a recent RG calculation \cite{Picco2}.
Moreover logarithmic corrections are not found in this region.
The results in Ref.~\cite{LR_SRnumeric} are compatible with those in Ref.~\cite{Picco}, due to the larger error bars.
Concluding, the scenario at the lower critical exponent is far from being clear.
The three proposed behaviours are summarized in Fig. \ref{Fig:eta}.

A somehow related problem is the identification of a $(d,\sigma) \leftrightarrow D$  relation that links the exponent $\sigma$
of the LR model in $d$ dimensions
with the dimension $D$ of an effective SR model with the same critical behavior.
Comparing the field theory of a LR model in one dimension ($d=1$) with that of a SR model above its upper critical dimension ($D>D_U)$,
the relation
\begin{equation} \label{Eq:sigma}
\sigma=\frac{2}{D}
\end{equation}
is found.
The upper critical dimension for the FM SR model, $D_U=4$,
thus corresponds to the upper critical exponent for the FM LR model $\sigma_U=1/2$.
In the same way, the upper critical dimension for the SG SR model, $D_U=6$,
 corresponds to the upper critical exponent for the SG LR model $\sigma_U=1/3$.
Moreover $\sigma=0$ corresponds to $D=\infty$, as one can expect. However, this relation has a problem.
In fact, the exponent for which there is no more a phase transition,
$\sigma_L=1$, will correspond to a lower critical dimension in SR models $D_L=2$.
But we know that the lower critical dimension for a SR model (with discrete degrees of freedom) is $D_L=1$.
This problem can be overcome modifying slightly the matching relation \cite{eta-D}:
\begin{equation} \label{Eq:sigma2}
\sigma=\frac{2-\eta_{SR}(D)}{D}
\end{equation}
and with scaling arguments the following relations between the critical exponents for LR in $d=1$ and SR models can be found \cite{eta-D_2012}:
\begin{equation}
\begin{aligned}
 &\nu_{LR}(\sigma)=D\nu_{SR}(D); \quad 2-\eta_{LR}(\sigma)=\sigma=\frac{2-\eta_{SR}(D)}{D};\\
 &\gamma_{LR}(\sigma)=\gamma_{SR}(D);\quad \omega_{LR}(\sigma)=\frac{\omega_{SR}(D)}{D}.
\label{Eq:D-sigma_exponents}
\end{aligned}
\end{equation}

Each relation among the four in Eq. (\ref{Eq:D-sigma_exponents}) defines a $(d,\sigma) \leftrightarrow D$  correspondence.
If SR and LR models are in the same universality class, then all the four correspondence in Eq. (\ref{Eq:D-sigma_exponents}) are equivalent.

Please note that the useful aspect of our definitions of the models is that all the $(d,\sigma) \leftrightarrow D$
relations are valid both for the FM and the SG versions of the models.

If one wants to test the exactness of the equivalence between a D-dimensional SR model and a one-dimensional LR model,
one has to simulate a LR system at a value of $\sigma$ that corresponds to $D$ following for example Eq. (\ref{Eq:sigma2}),
and verify if there is the correspondence between all the exponents as in Eq. (\ref{Eq:D-sigma_exponents}).
For the FM there is not a systematic study of the correspondence,
while, only during the writing of this work, this problem has been analyzed for SG in Ref.~\cite{eta-D_2012} for $D=3$ and $D=4$.
For $D=4$ the matching between LR and SR models seems very good. For $D=3$ the data are evenly compatible, however errors are bigger
and the answer is not definite.
Anyhow it is reasonable that the correspondence between SR and LR models becomes weaker approaching the lower critical dimension.
Indeed at the upper
critical dimension the field theory is exactly the same, while at the lower
critical dimension for the FM model we know that SR and LR models have even qualitatively different behaviours.
In fact the SR model has a $T=0$ transition, while the LR model has a KT transition.

Another LR model, widely used is the Dyson hierarchical model (HM) \cite{Dyson} (see \cite{Meurice} for a review).
It is a particular one-dimensional LR model, in which the Hamiltonian of $2^n$ spins can be constructed iteratively in the following way:
\begin{equation}
H_n(s_1,...,s_{2^n}) = H_{n-1}(s_1,...,s_{2^{n-1}}) +
+ H_{n-1}(s_{2^{n-1}+1},...,s_{2^n}) + c^n \sum_{i<j=1}^{2^n} J_{ij}\,s_i\,s_j\;.
\label{eq:Hindip}
\end{equation}
The intensity of the interactions decreases with the level $n$ by a factor $c=2^{-(\sigma+1)}$.
One expects the model to behave like the usual LR one, with the same exponent $\sigma$, because the decaying at large scales of the
coupling intensity is the same.
Indeed the model undergoes a standard second order phase transition if $2^{-1}>c>2^{-2}$ \cite{Dyson} (i.e. $0<\sigma<1$). For $2^{-1}>c>c_U=2^{-3/2}$
(i.e. $0<\sigma<\sigma_U=1/2$),
the Gaussian solution of the field theory associated to this model is the stable one
and the critical exponents are the mean-field ones as for usual LR systems.
Again, for $2^{-3/2}>c>c_L=2^{-2}$ (i.e. $1/2<\sigma<\sigma_L=1$) the exponents differ from the classical ones,
but nobody has checked if and how much they differ from the LR ones.
The first order term in the $\epsilon$-expansion of the two models is the same, while the second order one differs slightly,
the coefficients being 4.445 for the HM and 4.368 for the LR model \cite{nu_hier}.

One crucial difference between the two models is that for the borderline case $\sigma_L=1$ there is no KT phase-transition for the HM.
Indeed in the HM all the interactions are weaker than in the usual LR model.
For this reason if the HM has a transition, it implies that the LR model has a transition too, but the vice-versa is not necessarily true.
Nonetheless, there is a KT phase transition also in the HM for $\sigma=1$ if interactions at level $n$, $J_n=2^{-2 n}$, are made slightly stronger, i.e.\ $J_n=2^{-2 n}\log(n)$ \cite{LCD-log-Dyson}.

\section{New results on the connection between LR and SR models}

\subsection{Monte Carlo algorithm and data analysis}

To the best of our knowledge there exist no estimates of the critical exponents in $d=1$ for values of $\sigma$ corresponding to $D=2$ and $D=3$ following Eq.~(\ref{Eq:sigma2}) neither for the power law LR models nor for the HM.
For this reason we have performed Monte Carlo simulations at these values of $\sigma$.
Indeed Eq. (\ref{Eq:sigma2}) was introduced recently studying SG models, and was never applied to FM,
for which the relation (\ref{Eq:sigma}) was often used.
Moreover we want to see how similar is the HM with respect to the power law LR model.

We have simulated the $d=1$ LR model using the cluster algorithm proposed in Ref.~\cite{cluster_LR}, where the use of the cumulative probability distribution for adding a new spin to the cluster to be flipped allows to keep running times $O(N)$ even if the model is fully connected\footnote{Standard cluster algorithms usually require $O(N^2)$ operations for fully connected models.}.

In cluster algorithms, a first spin $\sigma_i$ is randomly chosen and the neighbors $\sigma_j$ having the same sign are inserted in the cluster with a probability
\begin{equation}
p_j=1-e^{-2\beta J_{ij}}.
\label{Eq:clusterprobability}
\end{equation}
The probability that the first neighbor to be included in the cluster is the $j$-th from the reference spin is
\begin{equation}
P(j)=p_j(1-p_{j-1})...(1-p_{1})\;.
\label{Eq:clusterP}
\end{equation}
Thus, by defining the cumulative bond probability
\begin{equation}
C(j)=\sum_{n=1}^jP(n)\;,
\label{Eq:cumulative}
\end{equation}
and extracting a random number $r$ uniformly in $[0,1]$, if $C(j-1)<r\leq C(j)$
then the first spin included in the cluster is the $j$-th. The condition on the spin being parallel
to those in the cluster is checked after the
selection. If the selected spin is antiparallel to those in the cluster, it is not added.

After the first neighbor has been chosen, we want to include in the cluster spins at distance $k>j$.
Eq.~(\ref{Eq:clusterP}) is generalized to
\begin{equation}
P_j(k)=p_k(1-p_{k-1})...(1-p_{j+1})\;,
\end{equation}
and it leads to a cumulative bond probability
\begin{equation}
C_j(k)=\sum_{n=j+1}^kP_j(n)=1-\exp\(({\sum_{n=j+1}^k-2\beta J_n}\))\;,
\label{Eq:cumulative2}
\end{equation}
where Eq.~(\ref{Eq:clusterprobability}) has been used to obtain the last expression
($J_n$ is the coupling between spins at distance $n$).
A new random number is extracted and a new spin is selected. Spins are added in this way until the maximum distance $N/2$ is reached.
Then we try to add neighbors starting from all the other spins already inserted in the cluster in the same way.
Naturally in this procedure we have to take into account that there are more than one spin at distance $k$
(especially in dimensions higher than 1),
and we must ensure that every spin is counted with the right probability.

Given $C_j(k)$ we construct a look-up table to calculate the distance $k$ associated to the random number extracted.
In this way the cumulative probability is calculated only once at the beginning and it is the same for all the spins,
since the system is homogeneous.
Moreover only $C(j)$ has to be computed, since $C_j(k)$ can be derived from it as
$$
C_j(k)=\frac{C(k)-C(j)}{1-C(j)}\;.
$$
Once the random number is extracted, we search in the look-up table to determine $k$. This operation has a cost $O(\log(N))$.
The main advantage of this method is that it is exact, at variance to the one of Ref.~\cite{cluster_LR}.

We have used periodic boundary conditions, such that two spins $i$ and $j$ interact
with a single coupling that depends on the minimum distance between them:
$r_{ij}=\min(|i-j|, L-|i-j| )$, where $L$ is the size of the system.

We have performed Monte Carlo simulations of at least $10^6$ MCS.
We have checked for the equilibration dividing the measurements in bins with a geometrically growing size, and we have assumed that the system has reached the equilibrium when the average of the magnetization in at least the last two bins is the same within the error (that is at least $3/4$ of the simulation is sampling the same average magnetization).
We have found that the equilibration time is $\tau \simeq 10^5$ cluster MC steps for the largest sizes.
Willing to compute the susceptibility and the Binder parameter, we need the second and fourth moments of the magnetization.
We have obtained two different estimates for these quantities. The first estimate is the usual one:
$$
m^2=\<\Big(\frac{1}{N}\sum_i \sigma_i\Big)^2\>\;,\qquad
m^4=\<\Big(\frac{1}{N}\sum_i \sigma_i\Big)^4\>\;.
$$
The second method uses the \textit{improved estimators} that can be defined when cluster algorithms are used \cite{improved_estimators}:
\begin{equation}
m^2=\frac{1}{N}\<|C|\>\;, \qquad
m^4=\frac{3}{N^2}\<|C||C'|\>-\frac{2}{N^3}\<|C|^3\>\;.
\label{Eq:improved}
\end{equation}
where $|C|$ and $|C'|$ are the sizes of flipped clusters.
Operatively, we compute $\<|C||C'|\>$ in the following way: we choose randomly a spin and, starting from it, we construct a cluster.
We call $|C|$ the number of spins of this first cluster. Then we choose a second spin randomly.
If it is in the cluster we already built, we put $|C'|=|C|$.
If it is not in that cluster, we construct a new cluster starting from it, and we call the new cluster size $|C'|$.
$C$ and $C'$ are always disjoint (i.e., non overlapping).
Please note that we can not compute the average $\<|C||C'|\>$ simply as $\<|C|^2\>$  because in this way we would not take into account the condition $|C|+|C'|\leq N$.

While the improved estimator for $m^2$ in Eq.~(\ref{Eq:improved}) has been already introduced in Ref.~\cite{improved_estimators}, we believe the one for $m^4$ is new.
We have computed the susceptibility and the Binder parameter and their errors with the jackknife method separately for the two methods.
At the end we have taken the weighted average between the two values.
In this way we are conscious that we are underestimating a little the error
because the two measures are correlated but we assume them to be uncorrelated when we perform the weighted average.

We have used this method to simulate a one-dimensional LR model with values of $\sigma$ corresponding to $D=2$ and $D=3$.
In $D=2$ we know exactly the exponent $\eta=\frac{1}{4}$ and it corresponds to $\sigma=\frac{2-1/4}{2}=0.875$.
In $D=3$, $\eta=0.0364(5)$ as found in Ref.~\cite{Pellissetto} and it corresponds to $\sigma=\frac{2-0.0364}{3}=0.65453$.
We have computed the critical exponents $\nu$ and $\omega$ using a Finite Size Scaling (FSS) analysis.

A great advantage of LR models is that the $\eta$ exponent is not renormalized in the non-mean-field region as explained before; thus we know its analytical expression: $\eta=2-\sigma$.
For this reason we can compute from the susceptibility $\chi=\frac{N}{T}\<m^2\>$ the scale-invariant quantity $\chi_L/L^{\sigma}$.
Another quantity that we look at is the dimensionless Binder parameter:
$B=\frac{1}{2}\Big[3-\frac{\langle m^4 \rangle}{\langle m^2 \rangle^2}\Big]$.
Both observables should cross at $T_c$ for large sizes.

\begin{figure}[t]
\begin{center}
\includegraphics[width=0.7\columnwidth]{chi_d1D3}
\includegraphics[width=0.7\columnwidth]{binder_d1D3}
\end{center}
\caption{\label{Fig:binder_chi} Scale invariant observable $\chi_L/L^{\sigma}$ (left) and Binder cumulant (right), computed at different $n=\log_2L$, as a function of the temperature $T$, at $\sigma=0.654533$.
The curves at different sizes should cross at a temperature that approaches $T_c$ when $L$ grows. The Binder cumulant shows stronger corrections to scaling with respect to $\chi_L/L^{\sigma}$.}
\end{figure}

\begin{figure}[t]
\begin{center}
\subfloat{\includegraphics[width=0.5\columnwidth]{omega_d1D3}}
\subfloat{\includegraphics[width=0.5\columnwidth]{nu_d1D3}}
\end{center}
\caption{\label{Fig:omega} Left: quotient of the Binder parameter for $L$ and $2L$ at $T^*_L$,
computed at the crossing temperature of $\chi_L/L^{\sigma}$.
The straight line is the best fit using Eq. (\ref{Eq:omega}) with $\omega$ left as a free parameter.
Right: quotient of the derivative of the Binder parameter at $T^*_L$.
The straight line is the best fit as a function of $L^{-\omega}$ using Eq. (\ref{Eq:nu}),
with $\omega$ determined from the previous fit and the intercept $2^{1/\nu}$ left as a free parameter.}
\end{figure}

In Fig. \ref{Fig:binder_chi} the two observables $\chi_L/L^{\sigma}$ and $B_L$ are plotted as a function of the temperature,
around the critical
temperature $T_c$, for different sizes $L=2^n$ of the systems.
We have extracted the temperatures $T^*_L$ of the crossing of $\chi_L/L^{\sigma}$ for sizes $L=2^n$ and $L'=2L=2^{n+1}$.
They should approach the critical point following:
\begin{equation}
T^*_L=T^*_{\infty}(1+aL^{-\omega-\frac{1}{\nu}}).
\label{Eq:T_c}
\end{equation}
We have computed the values of the Binder parameter $B(L,T^*_L)$ and the quotient $Q=\frac{B(2L,T^*_L)}{B(L,T^*_L)}$ at the previously extracted temperatures $T_L^*$.
The latter behaves as:
\begin{equation}Q=\frac{B(2L,T^*_L)}{B(L,T^*_L)}=1+bL^{-\omega}.
 \label{Eq:omega}
\end{equation}
Thus we have performed a fit with $\omega$ left as a free parameter.
The results are shown in the left side of Fig. \ref{Fig:omega}.
Once we have determined $\omega$, we extract the derivative of the Binder parameter at $T_L^*$, $B'(L,T^*_L)$, as the angular coefficient of the straight line passing through the data. We compute the quotient $Q=\frac{B'(2L,T^*_L)}{B'(L,T^*_L)}$ that follows:
\begin{equation}
 \frac{B'(2L,T^*_L)}{B'(L,T^*_L)}=2^{1/\nu}+cL^{-\omega}.
\label{Eq:nu}
\end{equation}
Thus, using the value of $\omega$ previously determined and performing a linear fit as a function of $L^{-\omega}$,
we extract the
value of $\nu$ from the intercept. The results are shown in the right panel of Fig. \ref{Fig:omega}.
At this point, fitting with a line the values of $T^*_L$ as a function of $L^{-\omega-1/\nu}$,
with the previously determined $\omega$ and $\nu$,
we can extract $T_c$ as the intercept, using Eq.~(\ref{Eq:T_c}).

\subsection{Results for the 1d LR models}

The results in $d=1$ for $\sigma=0.875$ (corresponding to $D=2$) and $\sigma=0.654533$ (corresponding to $D=3$) are the following:
$$\frac{1}{\nu_{LR}(0.875)}=0.4124(13), \quad T_c(0.875)=2.10589(1)$$
$$\frac{1}{\nu_{LR}(0.65453)}=0.506(14) ,\quad \omega_{LR}(0.65453)=0.201(11) ,\quad T_c(0.65453)=3.19289(2) .$$
For $\sigma=0.875$ it is quite impossible to determine $\omega$ because we see very little dependence of $T^*_L$ with $L$ and the quotient of the Binder parameter is nearly independent of the size.

For the HM, we performed the same analysis. The only difference is that we computed the Binder parameter
and the susceptibility exactly using the exact recursion relation for the probability of the magnetization at level $n$	:
\begin{equation*}
p_n(m) \propto e^{\beta c^n m^2}\!\sum_{m_L,m_R} p_{n-1}(m_L)\,p_{n-1}(m_R)\,\delta_{m_L+m_R,m}
\label{eq:Pm}
\end{equation*}
where $m_L$ and $m_R$ are the magnetizations of the half systems.
The results are:

$$\frac{1}{\nu_{HM}(0.875)}=0.3841(9),\quad \omega_{HM}(0.875)= 0.462(3)$$
$$\frac{1}{\nu_{HM}(0.65453)}=0.5186(72),\quad \omega_{HM}(0.65453)=0.212(5).$$
Naturally for the HM, many other methods can be used to obtain more precise results.
However, for the exponent $\omega$ only an estimate is available
\cite{omega-HM}
$$\omega_{HM}(2/3)=0.2185787,$$ consistent with our results.

The values for the $\nu$ and $\omega$ exponents of LR and HM for $\sigma=0.65453$ are in perfect agreement, while those for $\sigma=0.875$ differ.
Moreover if we compare them with the SR values \cite{Pellissetto}
$$\omega_{SR}(D=3)=0.84(4) , \quad \omega_{SR}(D=2)=2$$ and remembering the supposed relation between them,
$\omega_{SR}(D)=D\omega_{LR}(\sigma)$, it seems that LR models have bigger finite size effects (smaller $\omega$) than SR models.
Thus, looking at the $\omega$ exponent, Eq. (\ref{Eq:D-sigma_exponents}) is not satisfied, especially for $D=2$.

If we compare the values for the $\nu$ exponent with the SR ones,
$\nu_{SR}(2)=1$ and $\nu_{SR}(3)=0.6301(4)$ \cite{Pellissetto},
we see that Eq.~(\ref{Eq:D-sigma_exponents}) is a good approximation even for $D=3$ (that is near enough to the upper critical dimension)
 [$\nu_{LR}=1.976(55) \simeq 3 \nu_{SR} = 1.8903(12)$],
but it is no more good for $D=2$ [$\nu_{LR}=2.425(8) \neq 2 \nu_{SR} = 2$].

\subsection{Generalization of the $(d,\sigma) \leftrightarrow D$  relations in more than one dimension}

How can we generalize the $(d,\sigma) \leftrightarrow D$  relations if the LR model is defined in more than one dimension?
Let us first remark the notation: $d$ is the real dimension of the LR model while $D$ is the dimension of an equivalent SR model.
If we use the same arguments of the one-dimensional case in Ref.~\cite{eta-D_2012} for the scaling form of the free energy,
the relation
\begin{equation} \label{Eq:sigmaDd}
\frac{\sigma}{d}=\frac{2-\eta_{SR}(D)}{D}
\end{equation}
is obtained.

However Eq. (\ref{Eq:sigmaDd}) can be also obtained from another way.
In fact one can think that an approximate super-universality exists.
The conjecture is that the exponent $\gamma_{LR}(d,\sigma)$ and other quantities are approximately functions only of
$\hat{\sigma}=\sigma/d$.
This conjecture is exact in all the mean-field region. In fact, $\gamma_{LR}=1$ in the region $0<\hat{\sigma}<\frac{1}{2}$,
independently on $d$.
The SR model is recovered when $\sigma=\sigma_L(d)=2-\eta_{SR}(D=d)$ \cite{LR_SRnumeric,Sak}.
If now we use this information, we obtain that $\gamma_{SR}(D)=\gamma_{LR}(\hat{\sigma}=\frac{2-\eta_{SR}}{D})=\gamma_{LR}(\frac{\sigma}{d})$.
Thus the new relation between a LR model in $d$ dimensions and an effective SR model in $D$ dimensions is Eq. (\ref{Eq:sigmaDd}).
In this way we have connected two problems: the determination of the $(d,\sigma) \leftrightarrow D$  relation and the threshold $\sigma_L$
where the SR behaviour is recovered.
These problems are often viewed as disconnected, however we think that they are closely related.

Please note that the value of $\sigma_L$ is not universal:
$\sigma_L(1)=1, \sigma_L(2)=\frac{7}{4}$ \cite{Onsager}, $\sigma_L(3)=2-0.0364=1.9636$ \cite{Pellissetto}, $\sigma_L(4)=2$.
In the same way, $\hat{\sigma}=\hat{\sigma}_L$ is not universal:
$\hat{\sigma}_L(1)=1$ , $\hat{\sigma}_L(2)=0.875$ , $\hat{\sigma}_L(3)=0.65453$, $\hat{\sigma}_L(4)=0.5$.
For the exponent of the correlation length, using the scaling relation $\nu=\gamma/(2-\eta)$,
the known value of $\eta=2-\sigma$ in the LR region and Eq. (\ref{Eq:sigmaDd}),
one obtains:
\begin{equation} \label{Eq:nu-relation}
\nu_{SR}(D)=\frac{\gamma_{SR}(D)}{2-\eta_{SR}(D)}=\frac{d}{D\sigma}\gamma_{LR}\left(\frac{2-\eta_{SR}(D)}{D}\right)=\frac{d}{D}\nu_{LR}\left(\frac{2-\eta_{SR}(D)}{D}\right)
\end{equation}

In analogy with Eq. (\ref{Eq:D-sigma_exponents}), one can thus suppose that there exists a value of $\sigma$
that satisfies all the following relations for the critical exponents:

\begin{equation}
\begin{aligned}
 &d\,\nu_{LR}(\hat\sigma)=D\,\nu_{SR}(D); \quad \frac{2-\eta_{LR}(\hat\sigma)}{d}=\frac{2-\eta_{SR}(D)}{D};\\
 &\gamma_{LR}(\hat\sigma)=\gamma_{SR}(D);\quad \frac{\omega_{LR}(\hat\sigma)}{d}=\frac{\omega_{SR}(D)}{D}.
\label{Eq:Dd-sigma_exponents}
\end{aligned}
\end{equation}
Please note that the two dimensions $d$ and $D$ enter only through their ratio.

\subsection{Simulations in $d=2$}

Unfortunately there were not previous estimates for the $\nu$ exponent for the LR model in more than one dimension,
only during the completion of this work
in Ref.~\cite{Picco} the value $\nu_{LR}=0.96(2)$ for
$\sigma = 1.6$ in $d=2$ was reported, extracted from a Monte Carlo simulation.
For this reason we have also done simulations in $d=2$ to extract the exponents at values of $\sigma=1.20, 1.60$
and in particular at $\sigma=1.30906$
that corresponds to $D=3$ and $\sigma=1.75$ where the SR behaviour in $D=2$ should be recovered.
For $\sigma=1.2$ and $\sigma=1.30906$, the simulations have been performed with the same cluster algorithm
and the same analysis method
as for $d=1$.
The  obtained values for the $\nu_{LR}(\sigma)$ and $\omega_{LR}(\sigma)$ exponents and for the critical temperatures are:
$$\frac{1}{\nu_{LR}(1.2)}=1.024(34),\quad \omega_{LR}(1.2)=0.480(25) , \quad T_c(1.2)=6.83427(1),$$
$$\frac{1}{\nu_{LR}(1.30906)}= 1.014(33),\quad \omega_{LR}(1.30906)=0.32(15), \quad T_c(1.30906)=6.32546(4) .$$
The value of $\nu_{LR}$ at $\sigma=1.30906$ is compatible with the one for the $D=3$ SR model $\nu_{SR}=0.6301(4)$ \cite{Pellissetto}
following Eq. (\ref{Eq:Dd-sigma_exponents}):
$2 \nu_{LR} = 1.97(6) \simeq 3 \nu_{SR} = 1.8903(12)$.
The value of $\omega_{LR}$ at $\sigma=1.30906$ is very difficult to extrapolate because there are unusual non-monotonic finite size effects.

For $\sigma=1.6$ and $\sigma=1.75$ the finite size effects look extremely strong.
Indeed the size-dependent critical temperatures where the Binder cumulants cross drifts a lot by varying the system size, and it is not possible to extract the critical Binder value $B(\infty,T_c)$.
For this reason we move to use a slightly different model, where the sum over all images is made as in  \cite{LR_SRnumeric}, thus leading to new couplings
$$J_{ij}=\sum_{x=-\infty}^{\infty}\sum_{y=-\infty}^{\infty}\(((x_i-x_j+Lx)^2+(y_i-y_j+Ly)^2\))^{-(d+\sigma)/2}\;.$$
In the thermodynamic limit the two models (with and without images) are equivalent.
In Ref.~\cite{LR_SRnumeric} the authors were able to compute the contributions from all the images exactly, because they used slightly different couplings defined as
\begin{equation}
J_{ij}=\int_{|i-j|-\frac{1}{2}}^{|i-j|+\frac{1}{2}} x^{-(d+\sigma)} dx\;.
\label{Eq:coupling_integral}
\end{equation}
Since we use the original definition of the couplings, in principle it would not be possible to include all the images exactly.
To overcome this problem, we estimate the error that we commit on the largest coupling
(formally the one between two spins at distance 0) by including only the first $(2a)^2$ images (that is images within a distance $a$):
$$
\int_{\substack{|x|>a\\|y|>a}}\text{d}x\,\text{d}y\(((Lx)^2+(Ly)^2\))^{-\frac{2+\sigma}{2}}<2\pi L^{-(2+\sigma)}\int_a^{\infty}\text{d}r\, r^{-(1+\sigma)}=\frac{2\pi L^{-(2+\sigma)}}{\sigma a^\sigma},$$
and we choose $a$ such as to make this error smaller than $10^{-9}$.
At this point we compute the new couplings between any pair of spins as the sum of the couplings between the $(2a)^2$ images.
Due to the large values of $\sigma$, the number of images considered is always small. If $a$ results to be smaller than 10,
we choose $a=10$.
Adding the images, the observables show a reduced dependence on the system size, and the data analysis is cleaner.

We have not used the scale-invariant quantity $\chi_L/L^{2-\eta}$ because there is not agreement on the values of $\eta$ in this region. For this reason we have performed the following analysis.
We have looked at the temperatures $T^*_L$ at which the Binder cumulants for sizes $L$ and $2L$ cross.
These crossings scale according to Eq.~(\ref{Eq:T_c}). Then we have fitted the values of $B(L, T^*_L)$
with a power law function of the type
\begin{equation}
 B(L,T^*_L)\simeq B(\infty)+aL^{-\omega},
\label{Eq:scaling_omega}
\end{equation}
determining $\omega$. For $\sigma=1.75$, assuming that the Binder parameter at the critical point should recover the SR value, we have used the value of the Binder parameter $B_{\infty}(T_c) = 0.91588...$ \cite{Binder2D} in the fit to reduce the uncertainty in the determination of $\omega$.
At this point we have computed the quotient of the derivative of the Binder parameter at $T^*_L$ and extracted the exponent $\nu$. Knowing $\nu$ and $\omega$, we have estimated $T_c$.

The obtained results are the following:
$$\frac{1}{\nu_{LR}(1.6)}=0.996(33),\quad \omega_{LR}(1.6)=0.130(45), \quad T_c(1.6)=5.29321(4),$$
$$\frac{1}{\nu_{LR}(1.75)}=0.98(10),\quad \omega_{LR}(1.75)=0.213(8), \quad T_c(1.75)= 4.89455(17).$$
We do not observe any logarithmic corrections, as already noticed in Ref.~\cite{LR_SRnumeric}.
The value of $\omega_{LR}$ at $\sigma=1.75$ is much smaller than that for a SR model in $D=2$,
an explanation will be given in the following.
The value for $\nu_{LR}(1.75)$ is compatible with the SR one, $\nu_{SR}(D=2)=1$.

\subsection{At the lower critical $\sigma_L$}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=\columnwidth]{corr_s175}
\end{center}
\caption{The spin-spin correlation function for different sizes at $\sigma=1.75$ and $d=2$.
Left panel shows raw data, while in the right panel	boundary effects have been drastically reduced by plotting parametrically versus the variable $x'(x)\equiv\sin(\pi x/L)L/\pi$.}
\label{Fig:corr}
\end{figure}

In addition to the verification of the $(d,\sigma) \leftrightarrow D$  relations,
we have concentrated our attention to the problems arising when approaching
the value of $\sigma$ where the SR behaviour should be recovered.
In particular we want to verify whether the scenario of Ref.~\cite{Sak} holds, with
$\eta=\max(2-\sigma,\eta_{SR}=\frac{1}{4})$, or if, for $1.6 \leq\sigma\leq 2$,
the $\eta$ exponent interpolates smoothly between $2-\sigma$ and $\eta_{SR}$
as stated in Ref.~\cite{Picco}.
We notice that if the second scenario holds,
the superuniversality conjecture can not be verified in the region near to $\sigma_L(d)$
where the $\eta$ exponent interpolates smoothly between the two behaviours.
Superuniversality is compatible only with the first scenario.

We have tried to measure $\eta$ in $d=2$ at the lower critical value $\sigma=\sigma_L=1.75$.
We have performed MC simulations with a single image, because the use of images has the disadvantage that couplings slightly depend on the system size and consequently the small distance behavior of the correlation function does depend on the system size, making the study of finite size effects more complicated.
We have looked at the two-points correlation function at the critical point
that decays at large distances as $G(x)=\<\sigma(0)\sigma(x)\>=|x|^{-(d-2+\eta)}=|x|^{-\eta}$.
As it is customary, we have measured the correlation function along the principal axis:
$$
G(x)=\frac{1}{2N^2}\sum_{i,j}\Big(\<\sigma_{i,j}\sigma_{i+x,j}\>+\<\sigma_{i,j}\sigma_{i,j+x}\>\Big)\;.
$$
In the left panel of Fig.~\ref{Fig:corr} we plot the spin-spin correlation function for different sizes.
We notice that the effects due to the periodic boundary conditions, that actually imply the condition $G'(L/2)=0$, are rather severe and make hard to interpolate the data.
However the use of the variable $x'(x)\equiv\sin(\pi x/L)L/\pi$, that is actually an identity $x'(x)=x$ for $x\ll L$, is able to reduce drastically such boundary effects (see the right panel in Fig.~\ref{Fig:corr}). In the rest of the analysis we will use the rescaled variable $x'$, which is equal to $x$ in the thermodynamic limit, but allows for a better fitting of data at finite values of $L$.

The correlation function $G(x)$ at $\sigma=1.75$ can not be interpolated by a single power law: as shown in the right panel in Fig.~\ref{Fig:corr}, since $G(x)$ seems to decay faster at small distances and slower at large distances.
The same feature is not present at smaller $\sigma$, near to the upper critical value $\sigma_U=1$, nor in the SR model in $D=2$.
What we are observing is not a finite-size effect because it persists at large sizes.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.49\columnwidth]{corr_s175_scaled}
\includegraphics[width=0.49\columnwidth]{corr_s175_scaledPicco}
\end{center}
\caption{Data for the spin-spin correlation function $G(x)$ measured at $\sigma=1.75$ ad $d=2$, rescaled by the asymptotic power law $x'(x)^\eta$, in order to highlight the corrections to the asymptotic decay.
Left: $\eta=0.25$ and $\delta=0.3$. Right: $\eta_P=0.332$ and $\delta_P=0.5$.}
\label{Fig:corr_s175_scaled}
\end{figure}

Since the $G(x)$ shows a small, but clear, upward curvature in a log-log scale we have interpolated the data through the following function
\begin{equation}
G(x) = \frac{A+B\,x'(x)^{-\delta}}{x'(x)^\eta}\;,
\end{equation}
that uses the variable $x'(x)$ (that cancels most of the boundary effects) and includes a short distance correcting term $Bx^{-(\eta+\delta)}$ to the large distance power law decay $A x^{-\eta}$.
In Fig.~\ref{Fig:corr_s175_scaled} we plot $G(x)\,x'(x)^\eta$ versus the correcting term $x'(x)^{-\delta}$ and we observe a rather good linear behavior (the straight line is a linear fit to the $L=2048$ data).
In the left panel we have used $\eta=0.25$ and $\delta=0.3$, while on the right panel we have used the value for $\eta$ reported by Picco in Ref.~\cite{Picco}, that is $\eta_P=0.332$, and $\delta_P=0.5$ (please notice that the results are not very sensitive to the values of $\delta$ and $\delta_P$).

By looking to the data in Fig.~\ref{Fig:corr_s175_scaled} we make two observations. Firstly, the linearity of the data in both panels is similar, with a small preference to exponents used in the left panel: so we see no reason to use an $\eta$ exponent different from the theoretical prediction, $\eta=2-\sigma$. Secondly, a more careful inspection of the data reveals that using the exponent $\eta_P$ the data at large distances (small values of $x^{-\delta}$) always tend to bend up. A possible explanation for this observation is that the value $\eta_P$ for the exponent found by Picco is somehow a compromise between the asymptotic decay $x^{-\eta}$ and the pre-asymptotic correcting term $x^{-\eta-\delta}$: indeed data in the right panel of Fig.~\ref{Fig:corr_s175_scaled} show smaller corrections (they are flatter), but eventually tend to increase because the exponent $\eta_P$ is likely to be larger than the true exponent $\eta$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\columnwidth]{chis175fit}
\end{center}
\caption{\label{Fig:chifit} Log-log plot of the susceptibility at the maximum as a function of the size, for $\sigma=1.75$ and $d=2$.
Two fits using $f(x)=L^{2-\eta}(a+bL^{-\delta})$ and $g(x)=L^{2-\eta_P}(a+bL^{-\delta_P})$ are shown, which are both compatible with the data.}
\end{figure}

The effect of the two power-laws in $G(x)$ reflects also in the measure of the $\eta$ exponent from the susceptibility.
Indeed the susceptibility is the integral over $x$ of $G(x)$.
This means that if we measure $\chi$ as a function of the size of the system,
it will not follow a simple power law with exponent $2-\eta$.
Instead, it will be of the form:
\begin{equation}
\<\chi(L,T)\>=L^{2-\eta}\((a+bL^{-\delta}\))\[[F_{\chi}(L^{1/\nu}(T-T_c))+L^{-\omega}G_{\chi}(L^{1/\nu}(T-T_c))+\dots\]].
\label{Eq:FSSchi}
\end{equation}
The contribution proportional to $L^{2-\eta-\delta}$ is a new correcting term to the asymptotic behaviour, which is much bigger than the usual $L^{-\omega}$ correcting term.
Indeed, the correcting term $L^{-\omega}$ takes into account
the fact that the correlation function saturates and stops decaying at distances close to $L/2$
(as can be seen in the left panel of Fig.~\ref{Fig:corr}), but at these distance $G(x)$ is small and so it is also the correcting term $L^{-\omega}$.
On the contrary, the correcting term $L^{-\delta}$ is dominant at short distances, where the correlation $G(x)$ is large and this makes the correction $L^{-\delta}$ much larger than the $L^{-\omega}$ correcting term.

If this new correcting term is not properly taken into account, then the $\eta$ value is likely to be overestimated.
This may be the reason why in Ref.~\cite{Picco} the exponent $\eta$ is found to be bigger than the one predicted by the RG analysis.
The presence of this new correcting term can be also the reason why the
$\omega$ exponent found in the previous analysis is very small and not in agreement with the larger value of the SR model:
actually, we think that in the correction-to-scaling analysis we are measuring $\delta$ instead of $\omega$.

For each system size we have measured the connected susceptibility at its maximum,
which is a good proxy for the critical temperature.
Analogously to what we have done for the correlation function,
we have performed a fit to the maximum susceptibility as a function of the size, with the sum of two power laws: either with $f(x)=L^{2-\eta}(a+bL^{-\delta})$, where $\eta=2-\sigma$,
and with $g(x)=L^{2-\eta_P}(a+bL^{-\delta_P})$, where $\eta_P$ is the value reported by Picco in Ref.~\cite{Picco}.
We have ignored the corrections term $L^{-\omega}$ because, as discussed above, it is much smaller than the one considered.
The results are shown in Fig. \ref{Fig:chifit}.
The values of $\delta$ obtained are $\delta=0.41$ and $\delta_P=0.43$.
The values are similar to the ones obtained from the correlation function.
Again, both scenarios are compatible with the data and much larger sizes are needed to exclude one of the two.

The two power laws behaviour is a very strange feature of the correlation function,
because it is not present in the usual SR model, nor in the LR one far from the lower critical $\sigma$.
We leave for a future work to understand its physical origin and to eventually provide an analytical description of it.

\subsection{Check of the superuniversality conjecture}

\begin{figure}[t]
\includegraphics[width=\columnwidth]{nu}
\caption{\label{Fig:nu} $1/(\nu d)$ as a function of the exponent $\hat{\sigma}=\sigma/d$ for HM in $d=1$ and LR model in $d=1$ and $d=2$
as found in various works and in this work.
The SR values follow the matching formula (\ref{Eq:sigmaDd}).}
\end{figure}

At this point we want to verify the superuniversality conjecture or, equivalently, Eq. (\ref{Eq:sigmaDd}) and (\ref{Eq:nu-relation}).
For this reason we summarize the results for the critical exponents in the literature and in this work.
In Fig. \ref{Fig:nu}, $1/(d\nu)$ is plotted as a function of the parameter $\hat{\sigma}$ in the non mean-field region,
for the HM model as found in Ref.~\cite{nu_hier}, and for the LR one-dimensional model, from Ref.~\cite{nu_LR} and \cite{nu_LR2}.
For the LR $d=1$ and $d=2$ model, our results are reported too.
From Fig. \ref{Fig:nu} it is clear that the two analyzed one-dimensional models (HM and the LR one)
are not in the same universality class.
While their critical exponents are quite similar near to the upper critical $\sigma_U=1/2$,
the differences grow approaching the lower critical $\sigma_L=1$.
This is reasonable, because we know that the two models have very different behaviours at $\sigma_L=1$.

To verify the exactness of Eq. (\ref{Eq:nu-relation}),
in Fig. \ref{Fig:nu} the values of the exponent of the SR model as found in Ref.~\cite{Pellissetto}
are placed at the corresponding value of $\hat{\sigma}$ as in Eq. (\ref{Eq:sigmaDd}):
$\nu_{SR}(2)=1$ for $D=2$ corresponds to $\hat{\sigma}=0.875$, $\nu_{SR}(3)=0.6301(4)$
for $D=3$ corresponds to $\hat{\sigma}=0.65453$.
Eq. (\ref{Eq:nu-relation})
is a good approximation near to the upper critical dimension (it is good for $D=3$) but it is no more good for $D=2$.
Remembering also the results for the $\omega$ exponent, we can assert
that it is not possible to find a single value of $\sigma$
that verify the equivalence for all the critical exponents as in Eq. (\ref{Eq:Dd-sigma_exponents}).

The lines are the third order $\epsilon$-expansion for $d=2$ and $d=3$ as found in Ref.~\cite{fieldtheoryLR},
where the third order term has been fixed imposing that the curves recover the SR value at $\hat{\sigma}_L(d)$,
and the second order $\epsilon$-expansion for $d=1$.
For $d=1$ we have not fixed the third order because at $\sigma_L(1)=1$ there is not a second order phase transition.
For this reason the curve for $1/\nu$ as a function of $\sigma$ does not approach the point $\sigma_L(1)=1$ smoothly, but with a divergent derivative.
Our data for $d=2$ are in agreement with the $\epsilon$-expansion.

\begin{figure}[t]
\includegraphics[width=\columnwidth]{gamma}
\caption{\label{gamma} $1/\gamma$ as a function of the exponent $\hat{\sigma}=\sigma/d$
in the non mean-field region for HM in $d=1$ and LR model in $d=1$ and $d=2$ as found in various works and in this work.}
\end{figure}

In Fig. \ref{gamma}, $1/\gamma$ is plotted as a function of $\hat{\sigma}$ in the non mean-field region.
The super-universality conjecture is not exact but it is a good approximation near $\sigma_U$.
In fact $1/\gamma$ for $d=1$ and $d=2$
is nearly independent from $d$ and the two curves are near when plotted versus $\hat{\sigma}$.
The values for the SR model should be the end point of the line for $\gamma(\hat{\sigma})$ with $d=2$ and $d=3$, placed at $\hat{\sigma}_L(2)$ and $\hat{\sigma}_L(3)$.
The lines are the third order epsilon-expansion as found in Ref.~\cite{fieldtheoryLR},
where the third order has been fixed (as before) imposing that the curves recover the SR value at $\hat{\sigma}_L(d)$.

\section{Conclusions}

We have analyzed the connection between LR and SR systems.
For simplicity we have considered the ferromagnetic version of the models, given that the connection
we are interested in is still not well understood even in this simple case.
First of all we have analyzed the $d=1$ LR ferromagnetic model,
for which the couplings have a power-law decaying with exponent $\sigma$,
and we have compared it with a SR system in $D$ dimensions.
We have reviewed all the $(d,\sigma) \leftrightarrow D$  relations proposed in the literature and we have analyzed their accuracy
performing Monte Carlo simulations to measure the exponents of the LR model through finite size scaling.
We have compared them with the exponents of SR systems available in the literature.
We have found that near to the upper critical dimension a reliable $(d,\sigma) \leftrightarrow D$  relation exists:
it means that, for example, for $D=3$ a value of $\sigma$ exists for which all the exponents of the LR and SR models are very close,
while near to the lower critical dimension,
for example for $D=2$, it is not possible to find a value of $\sigma$ for which all the exponents of the LR model corresponds
to those of the SR one.

Then we have generalized the $(d,\sigma) \leftrightarrow D$  relation for LR systems in $d$ dimensions showing that the dimensions $D$
of the SR system and $d$ of the LR one enter only through their ratio $d/D$.
The $(d,\sigma) \leftrightarrow D$  relations in Eq. (\ref{Eq:Dd-sigma_exponents}) can be deduced also from a superuniversality conjecture.
We have verified this property performing Monte Carlo simulations at various values of $\sigma$ for $d=2$
to measure the critical exponents.
The superuniversality conjecture is a good approximation near to the upper critical dimension and
becomes worst going towards the lower critical one.

Finally for the $d=2$ LR model we have studied the region near to the lower critical dimension $\sigma_L=1.75$.
We have discovered that in this region the correlation function has a very strange behaviour,
characterized by two decaying power-laws.
This makes difficult to measure with high precision the critical exponents and the lower critical dimension.
Standard finite size scaling arguments do not help since the subdominant power law has an effect much larger than leading order finite size effects.
This kind of critical correlation function (with two different power laws)
can easily lead to overestimate $\eta$ if a proper fit with a double power-law is not performed.
Although we have performed the improved fit with two power laws, the exponent of the asymptotic decay,
i.e.\ the critical exponent $\eta$, has a very large uncertainty,
that makes it compatible both with the standard RG calculation by Sak \cite{Sak} and with the recent proposal by Picco \cite{Picco}.
According to Occam's razor, we see no reason to propose a different scenario \cite{Picco,Picco2}
as long as the numerical data, properly fitted, are compatible with the standard RG scenario proposed by Sak 40 years ago \cite{Sak}.

An important theoretical challenge is to understand analytically the origin of the two power laws
appearing in the critical correlation function close to the lower critical $\sigma$.

\begin{acknowledgments}
This research has received financial support from the European
Research Council (ERC) through grant agreement No. 247328 and from the
Italian Research Minister through the FIRB project No. RBFR086NN1.
\end{acknowledgments}