text:
You are currently browsing the category archive for the 'math.PR' category.
Let be a large natural number, and let be a matrix drawn from the Gaussian Unitary Ensemble (GUE), by which we mean that is a Hermitian matrix whose upper triangular entries are iid complex gaussians with mean zero and variance one, and whose diagonal entries are iid real gaussians with mean zero and variance one (and independent of the upper triangular entries). The eigenvalues are then real and almost surely distinct, and can be viewed as a random point process on the real line. One can then form the -point correlation functions for every , which can be defined by duality by requiring
for any test function . For GUE, which is a continuous matrix ensemble, one can also define for distinct as the unique quantity such that the probability that there is an eigenvalue in each of the intervals is in the limit .
As is well known, the GUE process is a determinantal point process, which means that -point correlation functions can be explicitly computed as
for some kernel ; explicitly, one has
where are the (normalised) Hermite polynomials; see this previous blog post for details.
Using the asymptotics of Hermite polynomials (which then give asymptotics for the kernel ), one can take a limit of a (suitably rescaled) sequence of GUE processes to obtain the Dyson sine process, which is a determinantal point process on the real line with correlation functions
where is the Dyson sine kernel
A bit more precisely, for any fixed bulk energy , the renormalised point processes converge in distribution in the vague topology to as , where is the semi-circular law density.
where is the Dyson sine kernel
A bit more precisely, for any fixed bulk energy , the renormalised point processes converge in distribution in the vague topology to as , where is the semi-circular law density.
On the other hand, an important feature of the GUE process is its stationarity (modulo rescaling) under Dyson Brownian motion
which describes the stochastic evolution of eigenvalues of a Hermitian matrix under independent Brownian motion of its entries, and is discussed in this previous blog post. To cut a long story short, this stationarity tells us that the self-similar -point correlation function
obeys the Dyson heat equation
(see Exercise 11 of the previously mentioned blog post). Note that vanishes to second order whenever two of the coincide, so there is no singularity on the right-hand side. Setting and using self-similarity, we can rewrite this equation in time-independent form as
One can then integrate out all but of these variables (after carefully justifying convergence) to obtain a system of equations for the -point correlation functions :
where the integral is interpreted in the principal value case. This system is an example of a BBGKY hierarchy.
If one carefully rescales and takes limits (say at the energy level , for simplicity), the left-hand side turns out to rescale to be a lower order term, and one ends up with a hierarchy for the Dyson sine process:
Informally, these equations show that the Dyson sine process is stationary with respect to the infinite Dyson Brownian motion
where are independent Brownian increments, and the sum is interpreted in a suitable principal value sense.
I recently set myself the exercise of deriving the identity (3) directly from the definition (1) of the Dyson sine process, without reference to GUE. This turns out to not be too difficult when done the right way (namely, by modifying the proof of Gaudin's lemma), although it did take me an entire day of work before I realised this, and I could not find it in the literature (though I suspect that many people in the field have privately performed this exercise in the past). In any case, I am recording the computation here, largely because I really don't want to have to do it again, but perhaps it will also be of interest to some readers.
One of the basic general problems in analytic number theory is to understand as much as possible the fluctuations of the Möbius function , defined as when is the product of distinct primes, and zero otherwise. For instance, as takes values in , we have the trivial bound
and the seemingly slight improvement
is already equivalent to the prime number theorem, as observed by Landau (see e.g. this previous blog post for a proof), while the much stronger (and still open) improvement
is already equivalent to the prime number theorem, as observed by Landau (see e.g. this previous blog post for a proof), while the much stronger (and still open) improvement
is equivalent to the notorious Riemann hypothesis.
There is a general Möbius pseudorandomness heuristic that suggests that the sign pattern behaves so randomly (or pseudorandomly) that one should expect a substantial amount of cancellation in sums that involve the sign fluctuation of the Möbius function in a nontrivial fashion, with the amount of cancellation present comparable to the amount that an analogous random sum would provide; cf. the probabilistic heuristic discussed in this recent blog post. There are a number of ways to make this heuristic precise. As already mentioned, the Riemann hypothesis can be considered one such manifestation of the heuristic. Another manifestation is the following old conjecture of Chowla:
Conjecture 1 (Chowla's conjecture) For any fixed integer and exponents , with at least one of the odd (so as not to completely destroy the sign cancellation), we have
Note that as for any , we can reduce to the case when the take values in here. When only one of the are odd, this is essentially the prime number theorem in arithmetic progressions (after some elementary sieving), but with two or more of the are odd, the problem becomes completely open. For instance, the estimate
is morally very close to the conjectured asymptotic
for the von Mangoldt function , where is the twin prime constant; this asymptotic in turn implies the twin prime conjecture. (To formally deduce estimates for von Mangoldt from estimates for Möbius, though, typically requires some better control on the error terms than , in particular gains of some power of are usually needed. See this previous blog post for more discussion.)
Remark 1 The Chowla conjecture resembles an assertion that, for chosen randomly and uniformly from to , the random variables become asymptotically independent of each other (in the probabilistic sense) as . However, this is not quite accurate, because some moments (namely those with all exponents even) have the "wrong" asymptotic value, leading to some unwanted correlation between the two variables. For instance, the events and have a strong correlation with each other, basically because they are both strongly correlated with the event of being divisible by . A more accurate interpretation of the Chowla conjecture is that the random variables are asymptotically conditionally independent of each other, after conditioning on the zero pattern ; thus, it is the sign of the Möbius function that fluctuates like random noise, rather than the zero pattern. (The situation is a bit cleaner if one works instead with the Liouville function instead of the Möbius function , as this function never vanishes, but we will stick to the traditional Möbius function formalism here.)
A more recent formulation of the Möbius randomness heuristic is the following conjecture of Sarnak. Given a bounded sequence , define the topological entropy of the sequence to be the least exponent with the property that for any fixed , and for going to infinity the set of can be covered by balls of radius . (If arises from a minimal topological dynamical system by , the above notion is equivalent to the usual notion of the topological entropy of a dynamical system.) For instance, if the sequence is a bit sequence (i.e. it takes values in ), then there are only -bit patterns that can appear as blocks of consecutive bits in this sequence. As a special case, a Turing machine with bounded memory that had access to a random number generator at the rate of one random bit produced every units of time, but otherwise evolved deterministically, would have an output sequence that had a topological entropy of at most . A bounded sequence is said to be deterministic if its topological entropy is zero. A typical example is a polynomial sequence such as for some fixed ; the -blocks of such polynomials sequence have covering numbers that only grow polynomially in , rather than exponentially, thus yielding the zero entropy. Unipotent flows, such as the horocycle flow on a compact hyperbolic surface, are another good source of deterministic sequences.
Conjecture 2 (Sarnak's conjecture) Let be a deterministic bounded sequence. Then
This conjecture in general is still quite far from being solved. However, special cases are known:
- For constant sequences, this is essentially the prime number theorem (1).
- For periodic sequences, this is essentially the prime number theorem in arithmetic progressions.
- For quasiperiodic sequences such as for some continuous , this follows from the work of Davenport.
- For nilsequences, this is a result of Ben Green and myself.
- For horocycle flows, this is a result of Bourgain, Sarnak, and Ziegler.
- For the Thue-Morse sequence, this is a result of Dartyge-Tenenbaum (with a stronger error term obtained by Maduit-Rivat). A subsequent result of Bourgain handles all bounded rank one sequences (though the Thue-Morse sequence is actually of rank two), and a related result of Green establishes asymptotic orthogonality of the Möbius function to bounded depth circuits, although such functions are not necessarily deterministic in nature.
- For the Rudin-Shapiro sequence, I sketched out an argument at this MathOverflow post.
- The Möbius function is known to itself be non-deterministic, because its square (i.e. the indicator of the square-free functions) is known to be non-deterministic (indeed, its topological entropy is ). (The corresponding question for the Liouville function , however, remains open, as the square has zero entropy.)
- In the converse direction, it is easy to construct sequences of arbitrarily small positive entropy that correlate with the Möbius function (a rather silly example is for some fixed large (squarefree) , which has topological entropy at most but clearly correlates with ).
See this survey of Sarnak for further discussion of this and related topics.
In this post I wanted to give a very nice argument of Sarnak that links the above two conjectures:
Proposition 3 The Chowla conjecture implies the Sarnak conjecture.
The argument does not use any number-theoretic properties of the Möbius function; one could replace in both conjectures by any other function from the natural numbers to and obtain the same implication. The argument consists of the following ingredients:
- To show that , it suffices to show that the expectation of the random variable , where is drawn uniformly at random from to , can be made arbitrary small by making large (and even larger).
- By the union bound and the zero topological entropy of , it suffices to show that for any bounded deterministic coefficients , the random variable concentrates with exponentially high probability.
- Finally, this exponentially high concentration can be achieved by the moment method, using a slight variant of the moment method proof of the large deviation estimates such as the Chernoff inequality or Hoeffding inequality (as discussed in this blog post).
As is often the case, though, while the "top-down" order of steps presented above is perhaps the clearest way to think conceptually about the argument, in order to present the argument formally it is more convenient to present the arguments in the reverse (or "bottom-up") order. This is the approach taken below the fold.
There has been a lot of recent interest in the abc conjecture, since the release a few weeks ago of the last of a series of papers by Shinichi Mochizuki which, as one of its major applications, claims to establish this conjecture. It's still far too early to judge whether this proof is likely to be correct or not (the entire argument encompasses several hundred pages of argument, mostly in the area of anabelian geometry, which very few mathematicians are expert in, to the extent that we still do not even have a full outline of the proof strategy yet), and I don't have anything substantial to add to the existing discussion around that conjecture. (But, for those that are interested, the Polymath wiki page on the ABC conjecture has collected most of the links to that discussion, and to various background materials.)
In the meantime, though, I thought I might give the standard probabilistic heuristic argument that explains why we expect the ABC conjecture to be true. The underlying heuristic is a common one, used throughout number theory, and it can be summarised as follows:
Heuristic 1 (Probabilistic heuristic) Even though number theory is a deterministic subject (one does not need to roll any dice to factorise a number, or figure out if a number is prime), one expects to get a good asymptotic prediction for the answers to many number-theoretic questions by pretending that various number-theoretic assertions (e.g. that a given number is prime) are probabilistic events (with a probability that can vary between and ) rather than deterministic events (that are either always true or always false). Furthermore:
- (Basic heuristic) If two or more of these heuristically probabilistic events have no obvious reason to be strongly correlated to each other, then we should expect them to behave as if they were (jointly) independent.
- (Advanced heuristic) If two or more of these heuristically probabilistic events have some obvious correlation between them, but no further correlations are suspected, then we should expect them to behave as if they were conditionally independent, relative to whatever data is causing the correlation.
This is, of course, an extremely vague and completely non-rigorous heuristic, requiring (among other things) a subjective and ad hoc determination of what an "obvious reason" is, but in practice it tends to give remarkably plausible predictions, some fraction of which can in fact be backed up by rigorous argument (although in many cases, the actual argument has almost nothing in common with the probabilistic heuristic). A famous special case of this heuristic is the Cramér random model for the primes, but this is not the only such instance for that heuristic.
To give the most precise predictions, one should use the advanced heuristic in Heuristic 1, but this can be somewhat complicated to execute, and so we shall focus instead on the predictions given by the basic heuristic (thus ignoring the presence of some number-theoretic correlations), which tends to give predictions that are quantitatively inaccurate but still reasonably good at the qualitative level.
Here is a basic "corollary" of Heuristic 1:
Heuristic 2 (Heuristic Borel-Cantelli) Suppose one has a sequence of number-theoretic statements, which we heuristically interpet as probabilistic events with probabilities . Suppose also that we know of no obvious reason for these events to have much of a correlation with each other. Then:
- If , we expect only finitely many of the statements to be true. (And if is much smaller than , we in fact expect none of the to be true.)
- If , we expect infinitely many of the statements to be true.
This heuristic is motivated both by the Borel-Cantelli lemma, and by the standard probabilistic computation that if one is given jointly independent, and genuinely probabilistic, events with , then one almost surely has an infinite number of the occuring.
Before we get to the ABC conjecture, let us give two simpler (and well known) demonstrations of these heuristics in action:
Example 1 (Twin prime conjecture) One can heuristically justify the twin prime conjecture as follows. Using the prime number theorem, one can heuristically assign a probability of to the event that any given large integer is prime. In particular, the probability that is prime will then be . Making the assumption that there are no strong correlations between these events, we are led to the prediction that the probability that and are simultaneously prime is . Since , the Borel-Cantelli heuristic then predicts that there should be infinitely many twin primes.
Note that the above argument is a bit too naive, because there are some non-trivial correlations between the primality of and the primality of . Most obviously, if is prime, this greatly increases the probability that is odd, which implies that is odd, which then elevates the probability that is prime. A bit more subtly, if is prime, then is likely to avoid the residue class , which means that avoids the residue class , which ends up decreasing the probability that is prime. However, there is a standard way to correct for these local correlations; see for instance in this previous blog post. As it turns out, these local correlations ultimately alter the prediction for the asymptotic density of twin primes by a constant factor (the twin prime constant), but do not affect the qualitative prediction of there being infinitely many twin primes.
Example 2 (Fermat's last theorem) Let us now heuristically count the number of solutions to for various and natural numbers (which we can reduce to be coprime if desired). We recast this (in the spirit of the ABC conjecture) as , where are powers. The number of powers up to any given number is about , so heuristically any given natural number has a probability about of being an power. If we make the naive assumption that (in the coprime case at least) there is no strong correlation between the events that is an power, is an power, and being an power, then for typical , the probability that are all simultaneously powers would then be . For fixed , the total number of solutions to the Fermat equation would then be predicted to be
(Strictly speaking, we need to restrict to the coprime case, but given that a positive density of pairs of integers are coprime, it should not affect the qualitative conclusion significantly if we now omit this restriction.) It might not be immediately obvious as to whether this sum converges or diverges, but (as is often the case with these sorts of unsigned sums) one can clarify the situation by dyadic decomposition. Suppose for instance that we consider the portion of the sum where lies between and . Then this portion of the sum can be controlled by
which simplifies to
Summing in , one thus expects infinitely many solutions for , only finitely many solutions for (indeed, a refinement of this argument shows that one expects only finitely many solutions even if one considers all at once), and a borderline prediction of there being a barely infinite number of solutions when . Here is of course a place where a naive application of the probabilistic heuristic breaks down; there is enough arithmetic structure in the equation that the naive probabilistic prediction ends up being an inaccurate model. Indeed, while this heuristic suggests that a typical homogeneous cubic should have a logarithmic number of integer solutions of a given height , it turns out that some homogeneous cubics (namely, those associated to elliptic curves of positive rank) end up with the bulk of these solutions, while other homogeneous cubics (including those associated to elliptic curves of zero rank, including the Fermat curve ) only get finitely many solutions. The reasons for this are subtle, but certainly the high degree of arithmetic structure present in an elliptic curve (starting with the elliptic curve group law which allows one to generate new solutions from old ones, and which also can be used to exclude solutions to via the method of descent) is a major contributing factor.
Below the fold, we apply similar heuristics to suggest the truth of the ABC conjecture.
Van Vu and I have just uploaded to the arXiv our paper "Random matrices: Universality of local spectral statistics of non-Hermitian matrices". The main result of this paper is a "Four Moment Theorem" that establishes universality for local spectral statistics of non-Hermitian matrices with independent entries, under the additional hypotheses that the entries of the matrix decay exponentially, and match moments with either the real or complex gaussian ensemble to fourth order. This is the non-Hermitian analogue of a long string of recent results establishing universality of local statistics in the Hermitian case (as discussed for instance in this recent survey of Van and myself, and also in several other places).
The complex case is somewhat easier to describe. Given a (non-Hermitian) random matrix ensemble of matrices, one can arbitrarily enumerate the (geometric) eigenvalues as , and one can then define the -point correlation functions to be the symmetric functions such that
In the case when is drawn from the complex gaussian ensemble, so that all the entries are independent complex gaussians of mean zero and variance one, it is a classical result of Ginibre that the asymptotics of near some point as and is fixed are given by the determinantal rule
for and
for and
for , where is the reproducing kernel
(There is also an asymptotic for the boundary case , but it is more complicated to state.) In particular, we see that for almost every , which is a manifestation of the well-known circular law for these matrices; but the circular law only captures the macroscopic structure of the spectrum, whereas the asymptotic (1) describes the microscopic structure.
Our first main result is that the asymptotic (1) for also holds (in the sense of vague convergence) when is a matrix whose entries are independent with mean zero, variance one, exponentially decaying tails, and which all match moments with the complex gaussian to fourth order. (Actually we prove a stronger result than this which is valid for all bounded and has more uniform bounds, but is a bit more technical to state.) An analogous result is also established for real gaussians (but now one has to separate the correlation function into components depending on how many eigenvalues are real and how many are strictly complex; also, the limiting distribution is more complicated, being described by Pfaffians rather than determinants). Among other things, this allows us to partially extend some known results on complex or real gaussian ensembles to more general ensembles. For instance, there is a central limit theorem of Rider which establishes a central limit theorem for the number of eigenvalues of a complex gaussian matrix in a mesoscopic disk; from our results, we can extend this central limit theorem to matrices that match the complex gaussian ensemble to fourth order, provided that the disk is small enough (for technical reasons, our error bounds are not strong enough to handle large disks). Similarly, extending some results of Edelman-Kostlan-Shub and of Forrester-Nagao, we can show that for a matrix matching the real gaussian ensemble to fourth order, the number of real eigenvalues is with probability for some absolute constant .
There are several steps involved in the proof. The first step is to apply the Girko Hermitisation trick to replace the problem of understanding the spectrum of a non-Hermitian matrix, with that of understanding the spectrum of various Hermitian matrices. The two identities that realise this trick are, firstly, Jensen's formula
that relates the local distribution of eigenvalues to the log-determinants , and secondly the elementary identity
that relates the log-determinants of to the log-determinants of the Hermitian matrices
The main difficulty is then to obtain concentration and universality results for the Hermitian log-determinants . This turns out to be a task that is analogous to the task of obtaining concentration for Wigner matrices (as we did in this recent paper), as well as central limit theorems for log-determinants of Wigner matrices (as we did in this other recent paper). In both of these papers, the main idea was to use the Four Moment Theorem for Wigner matrices (which can now be proven relatively easily by a combination of the local semi-circular law and resolvent swapping methods), combined with (in the latter paper) a central limit theorem for the gaussian unitary ensemble (GUE). This latter task was achieved by using the convenient Trotter normal form to tridiagonalise a GUE matrix, which has the effect of revealing the determinant of that matrix as the solution to a certain linear stochastic difference equation, and one can analyse the distribution of that solution via such tools as the martingale central limit theorem.
The matrices are somewhat more complicated than Wigner matrices (for instance, the semi-circular law must be replaced by a distorted Marchenko-Pastur law), but the same general strategy works to obtain concentration and universality for their log-determinants. The main new difficulty that arises is that the analogue of the Trotter norm for gaussian random matrices is not tridiagonal, but rather Hessenberg (i.e. upper-triangular except for the lower diagonal). This ultimately has the effect of expressing the relevant determinant as the solution to a nonlinear stochastic difference equation, which is a bit trickier to solve for. Fortunately, it turns out that one only needs good lower bounds on the solution, as one can use the second moment method to upper bound the determinant and hence the log-determinant (following a classical computation of Turan). This simplifies the analysis on the equation somewhat.
While this result is the first local universality result in the category of random matrices with independent entries, there are still two limitations to the result which one would like to remove. The first is the moment matching hypotheses on the matrix. Very recently, one of the ingredients of our paper, namely the local circular law, was proved without moment matching hypotheses by Bourgade, Yau, and Yin (provided one stays away from the edge of the spectrum); however, as of this time of writing the other main ingredient – the universality of the log-determinant – still requires moment matching. (The standard tool for obtaining universality without moment matching hypotheses is the heat flow method (and more specifically, the local relaxation flow method), but the analogue of Dyson Brownian motion in the non-Hermitian setting appears to be somewhat intractible, being a coupled flow on both the eigenvalues and eigenvectors rather than just on the eigenvalues alone.)
I've just uploaded to the arXiv my paper The asymptotic distribution of a single eigenvalue gap of a Wigner matrix, submitted to Probability Theory and Related Fields. This paper (like several of my previous papers) is concerned with the asymptotic distribution of the eigenvalues of a random Wigner matrix in the limit , with a particular focus on matrices drawn from the Gaussian Unitary Ensemble (GUE). This paper is focused on the bulk of the spectrum, i.e. to eigenvalues with for some fixed .
The location of an individual eigenvalue is by now quite well understood. If we normalise the entries of the matrix to have mean zero and variance , then in the asymptotic limit , the Wigner semicircle law tells us that with probability one has
where the classical location of the eigenvalue is given by the formula
and the semicircular distribution is given by the formula
Actually, one can improve the error term here from to for any (see this previous recent paper of Van and myself for more discussion of these sorts of estimates, sometimes known as eigenvalue rigidity estimates).
From the semicircle law (and the fundamental theorem of calculus), one expects the eigenvalue spacing to have an average size of . It is thus natural to introduce the normalised eigenvalue spacing
and ask what the distribution of is.
As mentioned previously, we will focus on the bulk case , and begin with the model case when is drawn from GUE. (In the edge case when is close to or to , the distribution is given by the famous Tracy-Widom law.) Here, the distribution was almost (but as we shall see, not quite) worked out by Gaudin and Mehta. By using the theory of determinantal processes, they were able to compute a quantity closely related to , namely the probability
that an interval near of length comparable to the expected eigenvalue spacing is devoid of eigenvalues. For in the bulk and fixed , they showed that this probability is equal to
that an interval near of length comparable to the expected eigenvalue spacing is devoid of eigenvalues. For in the bulk and fixed , they showed that this probability is equal to
where is the Dyson projection
to Fourier modes in , and is the Fredholm determinant. As shown by Jimbo, Miwa, Tetsuji, Mori, and Sato, this determinant can also be expressed in terms of a solution to a Painleve V ODE, though we will not need this fact here. In view of this asymptotic and some standard integration by parts manipulations, it becomes plausible to propose that will be asymptotically distributed according to the Gaudin-Mehta distribution , where
A reasonably accurate approximation for is given by the Wigner surmise , which was presciently proposed by Wigner as early as 1957; it is exact for but not in the asymptotic limit .
Unfortunately, when one tries to make this argument rigorous, one finds that the asymptotic for (1) does not control a single gap , but rather an ensemble of gaps , where is drawn from an interval of some moderate size (e.g. ); see for instance this paper of Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou for a more precise formalisation of this statement (which is phrased slightly differently, in which one samples all gaps inside a fixed window of spectrum, rather than inside a fixed range of eigenvalue indices ). (This result is stated for GUE, but can be extended to other Wigner ensembles by the Four Moment Theorem, at least if one assumes a moment matching condition; see this previous paper with Van Vu for details. The moment condition can in fact be removed, as was done in this subsequent paper with Erdos, Ramirez, Schlein, Vu, and Yau.)
The problem is that when one specifies a given window of spectrum such as , one cannot quite pin down in advance which eigenvalues are going to lie to the left or right of this window; even with the strongest eigenvalue rigidity results available, there is a natural uncertainty of or so in the index (as can be quantified quite precisely by this central limit theorem of Gustavsson).
The main difficulty here is that there could potentially be some strange coupling between the event (1) of an interval being devoid of eigenvalues, and the number of eigenvalues to the left of that interval. For instance, one could conceive of a possible scenario in which the interval in (1) tends to have many eigenvalues when is even, but very few when is odd. In this sort of situation, the gaps may have different behaviour for even than for odd , and such anomalies would not be picked up in the averaged statistics in which is allowed to range over some moderately large interval.
The main result of the current paper is that these anomalies do not actually occur, and that all of the eigenvalue gaps in the bulk are asymptotically governed by the Gaudin-Mehta law without the need for averaging in the parameter. Again, this is shown first for GUE, and then extended to other Wigner matrices obeying a matching moment condition using the Four Moment Theorem. (It is likely that the moment matching condition can be removed here, but I was unable to achieve this, despite all the recent advances in establishing universality of local spectral statistics for Wigner matrices, mainly because the universality results in the literature are more focused on specific energy levels than on specific eigenvalue indices . To make matters worse, in some cases universality is currently known only after an additional averaging in the energy parameter.)
The main task in the proof is to show that the random variable is largely decoupled from the event in (1) when is drawn from GUE. To do this we use some of the theory of determinantal processes, and in particular the nice fact that when one conditions a determinantal process to the event that a certain spatial region (such as an interval) contains no points of the process, then one obtains a new determinantal process (with a kernel that is closely related to the original kernel). The main task is then to obtain a sufficiently good control on the distance between the new determinantal kernel and the old one, which we do by some functional-analytic considerations involving the manipulation of norms of operators (and specifically, the operator norm, Hilbert-Schmidt norm, and nuclear norm). Amusingly, the Fredholm alternative makes a key appearance, as I end up having to invert a compact perturbation of the identity at one point (specifically, I need to invert , where is the Dyson projection and is an interval). As such, the bounds in my paper become ineffective, though I am sure that with more work one can invert this particular perturbation of the identity by hand, without the need to invoke the Fredholm alternative.
In the last three notes, we discussed the Bourgain-Gamburd expansion machine and two of its three ingredients, namely quasirandomness and product theorems, leaving only the non-concentration ingredient to discuss. We can summarise the results of the last three notes, in the case of fields of prime order, as the following theorem.
Theorem 1 (Non-concentration implies expansion in ) Let be a prime, let , and let be a symmetric set of elements in of cardinality not containing the identity. Write , and suppose that one has the non-concentration property
for some and some even integer . Then is a two-sided -expander for some depending only on .
Proof: From (1) we see that is not supported in any proper subgroup of , which implies that generates . The claim now follows from the Bourgain-Gamburd expansion machine (Theorem 2 of Notes 4), the product theorem (Theorem 1 of Notes 5), and quasirandomness (Exercise 8 of Notes 3).
Remark 1 The same argument also works if we replace by the field of order for some bounded . However, there is a difficulty in the regime when is unbounded, because the quasirandomness property becomes too weak for the Bourgain-Gamburd expansion machine to be directly applicable. On theother hand, the above type of theorem was generalised to the setting of cyclic groups with square-free by Varju, to arbitrary by Bourgain and Varju, and to more general algebraic groups than and square-free by Salehi Golsefidy and Varju. It may be that some modification of the proof techniques in these papers may also be able to handle the field case with unbounded .
It thus remains to construct tools that can establish the non-concentration property (1). The situation is particularly simple in , as we have a good understanding of the subgroups of that group. Indeed, from Theorem 14 from Notes 5, we obtain the following corollary to Theorem 1:
Corollary 2 (Non-concentration implies expansion in ) Let be a prime, and let be a symmetric set of elements in of cardinality not containing the identity. Write , and suppose that one has the non-concentration property
for some and some even integer , where ranges over all Borel subgroups of . Then, if is sufficiently large depending on , is a two-sided -expander for some depending only on .
It turns out (2) can be verified in many cases by exploiting the solvable nature of the Borel subgroups . We give two examples of this in these notes. The first result, due to Bourgain and Gamburd (with earlier partial results by Gamburd and by Shalom) generalises Selberg's expander construction to the case when generates a thin subgroup of :
Theorem 3 (Expansion in thin subgroups) Let be a symmetric subset of not containing the identity, and suppose that the group generated by is not virtually solvable. Then as ranges over all sufficiently large primes, the Cayley graphs form a two-sided expander family, where is the usual projection.
Remark 2 One corollary of Theorem 3 (or of the non-concentration estimate (3) below) is that generates for all sufficiently large , if is not virtually solvable. This is a special case of a much more general result, known as the strong approximation theorem, although this is certainly not the most direct way to prove such a theorem. Conversely, the strong approximation property is used in generalisations of this result to higher rank groups than .
Exercise 1 In the converse direction, if is virtually solvable, show that for sufficiently large , fails to generate . (Hint: use Theorem 14 from Notes 5 to prevent from having bounded index solvable subgroups.)
Exercise 2 (Lubotzsky's 1-2-3 problem) Let .
- (i) Show that generates a free subgroup of . (Hint: use a ping-pong argument, as in Exercise 23 of Notes 2.)
- (ii) Show that if are two distinct elements of the sector , then there os no element for which . (Hint: this is another ping-pong argument.) Conclude that has infinite index in . (Contrast this with the situation in which the coefficients in are replaced by or , in which case is either all of , or a finite index subgroup, as demonstrated in Exercise 23 of Notes 2).
- (iii) Show that for sufficiently large primes form a two-sided expander family.
Remark 3 Theorem 3 has been generalised to arbitrary linear groups, and with replaced by for square-free ; see this paper of Salehi Golsefidy and Varju. In this more general setting, the condition of virtual solvability must be replaced by the condition that the connected component of the Zariski closure of is perfect. An effective version of Theorem 3 (with completely explicit constants) was recently obtained by Kowalski.
The second example concerns Cayley graphs constructed using random elements of .
Theorem 4 (Random generators expand) Let be a prime, and let be two elements of chosen uniformly at random. Then with probability , is a two-sided -expander for some absolute constant .
Remark 4 As with Theorem 3, Theorem 4 has also been extended to a number of other groups, such as the Suzuki groups (in this paper of Breuillard, Green, and Tao), and more generally to finite simple groups of Lie type of bounded rank (in forthcoming work of Breuillard, Green, Guralnick, and Tao). There are a number of other constructions of expanding Cayley graphs in such groups (and in other interesting groups, such as the alternating groups) beyond those discussed in these notes; see this recent survey of Lubotzky for further discussion. It has been conjectured by Lubotzky and Weiss that any pair of (say) that generates the group, is a two-sided -expander for an absolute constant : in the case of , this has been established for a density one set of primes by Breuillard and Gamburd.
— 1. Expansion in thin subgroups —
We now prove Theorem 3. The first observation is that the expansion property is monotone in the group :
Exercise 3 Let be symmetric subsets of not containing the identity, such that . Suppose that is a two-sided expander family for sufficiently large primes . Show that is also a two-sided expander family.
As a consequence, Theorem 3 follows from the following two statments:
Theorem 5 (Tits alternative) Let be a group. Then exactly one of the following statements holds:
- (i) is virtually solvable.
- (ii) contains a copy of the free group of two generators as a subgroup.
Theorem 6 (Expansion in free groups) Let be generators of a free subgroup of . Then as ranges over all sufficiently large primes, the Cayley graphs form a two-sided expander family.
Theorem 5 is a special case of the famous Tits alternative, which among other things allows one to replace by for any and any field of characteristic zero (and fields of positive characteristic are also allowed, if one adds the requirement that be finitely generated). We will not prove the full Tits alternative here, but instead just give an ad hoc proof of the special case in Theorem 5 in the following exercise.
Exercise 4 Given any matrix , the singular values are and , and we can apply the singular value decomposition to decompose
where and are orthonormal bases. (When , these bases are uniquely determined up to phase rotation.) We let be the projection of to the projective complex plane, and similarly define .
Let be a subgroup of . Call a pair a limit point of if there exists a sequence with and .
- (i) Show that if is infinite, then there is at least one limit point.
- (ii) Show that if is a limit point, then so is .
- (iii) Show that if there are two limit points with , then there exist that generate a free group. (Hint: Choose close to and close to , and consider the action of and on , and specifically on small neighbourhoods of , and set up a ping-pong type situation.)
- (iv) Show that if is hyperbolic (i.e. it has an eigenvalue greater than 1), with eigenvectors , then the projectivisations of form a limit point. Similarly, if is regular parabolic (i.e. it has an eigenvalue at 1, but is not the identity) with eigenvector , show that is a limit point.
- (v) Show that if has no free subgroup of two generators, then all hyperbolic and regular parabolic elements of have a common eigenvector. Conclude that all such elements lie in a solvable subgroup of .
- (vi) Show that if an element is neither hyperbolic nor regular parabolic, and is not a multiple of the identity, then is conjugate to a rotation by (in particular, ).
- (vii) Establish Theorem 5. (Hint: show that two square roots of in cannot multiply to another square root of .)
Now we prove Theorem 6. Let be a free subgroup of generated by two generators . Let be the probability measure generating a random walk on , thus is the corresponding generator on . By Corollary 2, it thus suffices to show that
for all sufficiently large , some absolute constant , and some even (depending on , of course), where ranges over Borel subgroups.
for all sufficiently large , some absolute constant , and some even (depending on , of course), where ranges over Borel subgroups.
As is a homomorphism, one has and so it suffices to show that
To deal with the supremum here, we will use an argument of Bourgain and Gamburd, taking advantage of the fact that all Borel groups of obey a common group law, the point being that free groups such as obey such laws only very rarely. More precisely, we use the fact that the Borel groups are solvable of derived length two; in particular we have
for all . Now, is supported on matrices in whose coefficients have size (where we allow the implied constants to depend on the choice of generators ), and so is supported on matrices in whose coefficients also have size . If is less than a sufficiently small multiple of , these coefficients are then less than (say). As such, if lie in the support of and their projections obey the word law (4) in , then the original matrices obey the word law (4) in . (This lifting of identities from the characteristic setting of to the characteristic setting of is a simple example of the "Lefschetz principle".)
for all . Now, is supported on matrices in whose coefficients have size (where we allow the implied constants to depend on the choice of generators ), and so is supported on matrices in whose coefficients also have size . If is less than a sufficiently small multiple of , these coefficients are then less than (say). As such, if lie in the support of and their projections obey the word law (4) in , then the original matrices obey the word law (4) in . (This lifting of identities from the characteristic setting of to the characteristic setting of is a simple example of the "Lefschetz principle".)
To summarise, if we let be the set of all elements of that lie in the support of , then (4) holds for all . This severely limits the size of to only be of polynomial size, rather than exponential size:
Proposition 7 Let be a subset of the support of (thus, consists of words in of length ) such that the law (4) holds for all . Then .
The proof of this proposition is laid out in the exercise below.
Exercise 5 Let be a free group generated by two generators . Let be the set of all words of length at most in .
- (i) Show that if commute, then lie in the same cyclic group, thus for some and .
- (ii) Show that if , there are at most elements of that commute with .
- (iii) Show that if , there are at most elements of with .
- (iv) Prove Proposition 7.
Now we can conclude the proof of Theorem 3:
Exercise 6 Let be a free group generated by two generators .
- (i) Show that for some absolute constant . (For much more precise information on , see this paper of Kesten.)
- (ii) Conclude the proof of Theorem 3.
— 2. Random generators expand —
We now prove Theorem 4. Let be the free group on two formal generators , and let be the generator of the random walk. For any word and any in a group , let be the element of formed by substituting for respectively in the word ; thus can be viewed as a map for any group . Observe that if is drawn randomly using the distribution , and , then is distributed according to the law , where . Applying Corollary 2, it suffices to show that whenever is a large prime and are chosen uniformly and independently at random from , that with probability , one has
for some absolute constant , where ranges over all Borel subgroups of and is drawn from the law for some even natural number .
for some absolute constant , where ranges over all Borel subgroups of and is drawn from the law for some even natural number .
Let denote the words in of length at most . We may use the law (4) to obtain good bound on the supremum in (5) assuming a certain non-degeneracy property of the word evaluations :
Exercise 7 Let be a natural number, and suppose that is such that for . Show that
for some absolute constant , where is drawn from the law . (Hint: use (4) and the hypothesis to lift the problem up to , at which point one can use Proposition 7 and Exercise 6.)
In view of this exercise, it suffices to show that with probability , one has for all for some comparable to a small multiple of . As has elements, it thus suffices by the union bound to show that
for some absolute constant , and any of length less than for some sufficiently small absolute constant .
for some absolute constant , and any of length less than for some sufficiently small absolute constant .
Let us now fix a non-identity word of length less than , and consider as a function from to for an arbitrary field . We can identify with the set . A routine induction then shows that the expression is then a polynomial in the eight variables of degree and coefficients which are integers of size . Let us then make the additional restriction to the case , in which case we can write and . Then is now a rational function of whose numerator is a polynomial of degree and coefficients of size , and the denominator is a monomial of of degree .
We then specialise this rational function to the field . It is conceivable that when one does so, the rational function collapses to the constant polynomial , thus for all with . (For instance, this would be the case if , by Lagrange's theorem, if it were not for the fact that is far too large here.) But suppose that this rational function does not collapse to the constant rational function. Applying the Schwarz-Zippel lemma (Exercise 23 from Notes 5), we then see that the set of pairs with and is at most ; adding in the and cases, one still obtains a bound of , which is acceptable since and . Thus, the only remaining case to consider is when the rational function is identically on with .
Now we perform another "Lefschetz principle" maneuvre to change the underlying field. Recall that the denominator of rational function is monomial in , and the numerator has coefficients of size . If is less than for a sufficiently small , we conclude in particular (for large enough) that the coefficients all have magnitude less than . As such, the only way that this function can be identically on is if it is identically on for all with , and hence for or also by taking Zariski closures.
On the other hand, we know that for some choices of , e.g. , contains a copy of the free group on two generators (see e.g. Exercise 23 of Notes 2). As such, it is not possible for any non-identity word to be identically trivial on . Thus this case cannot actually occur, completing the proof of (6) and hence of Theorem 4.
Remark 5 We see from the above argument that the existence of subgroups of an algebraic group with good "independence" properties – such as that of generating a free group – can be useful in studying the expansion properties of that algebraic group, even if the field of interest in the latter is distinct from that of the former. For more complicated algebraic groups than , in which laws such as (4) are not always available, it turns out to be useful to place further properties on the subgroup , for instance by requiring that all non-abelian subgroups of that group be Zariski dense (a property which has been called strong density), as this turns out to be useful for preventing random walks from concentrating in proper algebraic subgroups. See this paper of Breuillard, Guralnick, Green and Tao for constructions of strongly dense free subgroups of algebraic groups and further discussion.
Van Vu and I have just uploaded to the arXiv our paper "Random matrices: The Universality phenomenon for Wigner ensembles". This survey is a longer version (58 pages) of a previous short survey we wrote up a few months ago. The survey focuses on recent progress in understanding the universality phenomenon for Hermitian Wigner ensembles, of which the Gaussian Unitary Ensemble (GUE) is the most well known. The one-sentence summary of this progress is that many of the asymptotic spectral statistics (e.g. correlation functions, eigenvalue gaps, determinants, etc.) that were previously known for GUE matrices, are now known for very large classes of Wigner ensembles as well. There are however a wide variety of results of this type, due to the large number of interesting spectral statistics, the varying hypotheses placed on the ensemble, and the different modes of convergence studied, and it is difficult to isolate a single such result currently as the definitive universality result. (In particular, there is at present a tradeoff between generality of ensemble and strength of convergence; the universality results that are available for the most general classes of ensemble are only presently able to demonstrate a rather weak sense of convergence to the universal distribution (involving an additional averaging in the energy parameter), which limits the applicability of such results to a number of interesting questions in which energy averaging is not permissible, such as the study of the least singular value of a Wigner matrix, or of related quantities such as the condition number or determinant. But it is conceivable that this tradeoff is a temporary phenomenon and may be eliminated by future work in this area; in the case of Hermitian matrices whose entries have the same second moments as that of the GUE ensemble, for instance, the need for energy averaging has already been removed.)
Nevertheless, throughout the family of results that have been obtained recently, there are two main methods which have been fundamental to almost all of the recent progress in extending from special ensembles such as GUE to general ensembles. The first method, developed extensively by Erdos, Schlein, Yau, Yin, and others (and building on an initial breakthrough by Johansson), is the heat flow method, which exploits the rapid convergence to equilibrium of the spectral statistics of matrices undergoing Dyson-type flows towards GUE. (An important aspect to this method is the ability to accelerate the convergence to equilibrium by localising the Hamiltonian, in order to eliminate the slowest modes of the flow; this refinement of the method is known as the "local relaxation flow" method. Unfortunately, the translation mode is not accelerated by this process, which is the principal reason why results obtained by pure heat flow methods still require an energy averaging in the final conclusion; it would of interest to find a way around this difficulty.) The other method, which goes all the way back to Lindeberg in his classical proof of the central limit theorem, and which was introduced to random matrix theory by Chatterjee and then developed for the universality problem by Van Vu and myself, is the swapping method, which is based on the observation that spectral statistics of Wigner matrices tend to be stable if one replaces just one or two entries of the matrix with another distribution, with the stability of the swapping process becoming stronger if one assumes that the old and new entries have many matching moments. The main formalisations of this observation are known as four moment theorems, because they require four matching moments between the entries, although there are some variant three moment theorems and two moment theorems in the literature as well. Our initial four moment theorems were focused on individual eigenvalues (and later also to eigenvectors), but it was later observed by Erdos, Yau, and Yin that simpler four moment theorems could also be established for aggregate spectral statistics, such as the coefficients of the Greens function, and Knowles and Yin also subsequently observed that these latter theorems could be used to recover a four moment theorem for eigenvalues and eigenvectors, giving an alternate approach to proving such theorems.
Interestingly, it seems that the heat flow and swapping methods are complementary to each other; the heat flow methods are good at removing moment hypotheses on the coefficients, while the swapping methods are good at removing regularity hypotheses. To handle general ensembles with minimal moment or regularity hypotheses, it is thus necessary to combine the two methods (though perhaps in the future a third method, or a unification of the two existing methods, might emerge).
Besides the heat flow and swapping methods, there are also a number of other basic tools that are also needed in these results, such as local semicircle laws and eigenvalue rigidity, which are also discussed in the survey. We also survey how universality has been established for wide variety of spectral statistics; the -point correlation functions are the most well known of these statistics, but they do not tell the whole story (particularly if one can only control these functions after an averaging in the energy), and there are a number of other statistics, such as eigenvalue counting functions, determinants, or spectral gaps, for which the above methods can be applied.
In order to prevent the survey from becoming too enormous, we decided to restrict attention to Hermitian matrix ensembles, whose entries off the diagonal are identically distributed, as this is the case in which the strongest results are available. There are several results that are applicable to more general ensembles than these which are briefly mentioned in the survey, but they are not covered in detail.
We plan to submit this survey eventually to the proceedings of a workshop on random matrix theory, and will continue to update the references on the arXiv version until the time comes to actually submit the paper.
Finally, in the survey we issue some errata for previous papers of Van and myself in this area, mostly centering around the three moment theorem (a variant of the more widely used four moment theorem), for which the original proof of Van and myself was incomplete. (Fortunately, as the three moment theorem had many fewer applications than the four moment theorem, and most of the applications that it did have ended up being superseded by subsequent papers, the actual impact of this issue was limited, but still an erratum is in order.)
Van Vu and I have just uploaded to the arXiv our paper Random matrices: Sharp concentration of eigenvalues, submitted to the Electronic Journal of Probability. As with many of our previous papers, this paper is concerned with the distribution of the eigenvalues of a random Wigner matrix (such as a matrix drawn from the Gaussian Unitary Ensemble (GUE) or Gaussian Orthogonal Ensemble (GOE)). To simplify the discussion we shall mostly restrict attention to the bulk of the spectrum, i.e. to eigenvalues with for some fixed , although analogues of most of the results below have also been obtained at the edge of the spectrum.
If we normalise the entries of the matrix to have mean zero and variance , then in the asymptotic limit , we have the Wigner semicircle law, which asserts that the eigenvalues are asymptotically distributed according to the semicircular distribution , where
An essentially equivalent way of saying this is that for large , we expect the eigenvalue of to stay close to the classical location , defined by the formula
In particular, from the Wigner semicircle law it can be shown that asymptotically almost surely, one has
for all .
for all .
In the modern study of the spectrum of Wigner matrices (and in particular as a key tool in establishing universality results), it has become of interest to improve the error term in (1) as much as possible. A typical early result in this direction was by Bai, who used the Stieltjes transform method to obtain polynomial convergence rates of the shape for some absolute constant ; see also the subsequent papers of Alon-Krivelevich-Vu and of of Meckes, who were able to obtain such convergence rates (with exponentially high probability) by using concentration of measure tools, such as Talagrand's inequality. On the other hand, in the case of the GUE ensemble it is known (by this paper of Gustavsson) that has variance comparable to in the bulk, so that the optimal error term in (1) should be about . (One may think that if one wanted bounds on (1) that were uniform in , one would need to enlarge the error term further, but this does not appear to be the case, due to strong correlations between the ; note for instance this recent result of Ben Arous and Bourgarde that the largest gap between eigenvalues in the bulk is typically of order .)
A significant advance in this direction was achieved by Erdos, Schlein, and Yau in a series of papers where they used a combination of Stieltjes transform and concentration of measure methods to obtain local semicircle laws which showed, among other things, that one had asymptotics of the form
with exponentially high probability for intervals in the bulk that were as short as for some , where is the number of eigenvalues. These asymptotics are consistent with a good error term in (1), and are already sufficient for many applications, but do not quite imply a strong concentration result for individual eigenvalues (basically because they do not preclude long-range or "secular" shifts in the spectrum that involve large blocks of eigenvalues at mesoscopic scales). Nevertheless, this was rectified in a subsequent paper of Erdos, Yau, and Yin, which roughly speaking obtained a bound of the form
in the bulk with exponentially high probability, for Wigner matrices obeying some exponential decay conditions on the entries. This was achieved by a rather delicate high moment calculation, in which the contribution of the diagonal entries of the resolvent (whose average forms the Stieltjes transform) was shown to mostly cancel each other out.
As the GUE computations show, this concentration result is sharp up to the quasilogarithmic factor . The main result of this paper is to improve the concentration result to one more in line with the GUE case, namely
with exponentially high probability (see the paper for a more precise statement of results). The one catch is that an additional hypothesis is required, namely that the entries of the Wigner matrix have vanishing third moment. We also obtain similar results for the edge of the spectrum (but with a different scaling).
Our arguments are rather different from those of Erdos, Yau, and Yin, and thus provide an alternate approach to establishing eigenvalue concentration. The main tool is the Lindeberg exchange strategy, which is also used to prove the Four Moment Theorem (although we do not directly invoke the Four Moment Theorem in our analysis). The main novelty is that this exchange strategy is now used to establish large deviation estimates (i.e. exponentially small tail probabilities) rather than universality of the limiting distribution. Roughly speaking, the basic point is as follows. The Lindeberg exchange strategy seeks to compare a function of many independent random variables with the same function of a different set of random variables (which match moments with the original set of variables to some order, such as to second or fourth order) by exchanging the random variables one at a time. Typically, one tries to upper bound expressions such as
for various smooth test functions , by performing a Taylor expansion in the variable being swapped and taking advantage of the matching moment hypotheses. In previous implementations of this strategy, was a bounded test function, which allowed one to get control of the bulk of the distribution of , and in particular in controlling probabilities such as
for various thresholds and , but did not give good control on the tail as the error terms tended to be polynomially decaying in rather than exponentially decaying. However, it turns out that one can modify the exchange strategy to deal with moments such as
for various moderately large (e.g. of size comparable to ), obtaining results such as
after performing all the relevant exchanges. As such, one can then use large deviation estimates on to deduce large deviation estimates on .
In this paper we also take advantage of a simplification, first noted by Erdos, Yau, and Yin, that Four Moment Theorems become somewhat easier to prove if one works with resolvents (and the closely related Stieltjes transform ) rather than with individual eigenvalues, as the Taylor expansion of resolvents are very simple (essentially being a Neumann series). The relationship between the Stieltjes transform and the location of individual eigenvalues can be seen by taking advantage of the identity
for any energy level , which can be verified from elementary calculus. (In practice, we would truncate near zero and near infinity to avoid some divergences, but this is a minor technicality.) As such, a concentration result for the Stieltjes transform can be used to establish an analogous concentration result for the eigenvalue counting functions , which in turn can be used to deduce concentration results for individual eigenvalues by some basic combinatorial manipulations.
Van Vu and I have just uploaded to the arXiv our short survey article, "Random matrices: The Four Moment Theorem for Wigner ensembles", submitted to the MSRI book series, as part of the proceedings on the MSRI semester program on random matrix theory from last year. This is a highly condensed version (at 17 pages) of a much longer survey (currently at about 48 pages, though not completely finished) that we are currently working on, devoted to the recent advances in understanding the universality phenomenon for spectral statistics of Wigner matrices. In this abridged version of the survey, we focus on a key tool in the subject, namely the Four Moment Theorem which roughly speaking asserts that the statistics of a Wigner matrix depend only on the first four moments of the entries. We give a sketch of proof of this theorem, and two sample applications: a central limit theorem for individual eigenvalues of a Wigner matrix (extending a result of Gustavsson in the case of GUE), and the verification of a conjecture of Wigner, Dyson, and Mehta on the universality of the asymptotic k-point correlation functions even for discrete ensembles (provided that we interpret convergence in the vague topology sense).
For reasons of space, this paper is very far from an exhaustive survey even of the narrow topic of universality for Wigner matrices, but should hopefully be an accessible entry point into the subject nevertheless.
Van Vu and I have just uploaded to the arXiv our paper A central limit theorem for the determinant of a Wigner matrix, submitted to Adv. Math.. It studies the asymptotic distribution of the determinant of a random Wigner matrix (such as a matrix drawn from the Gaussian Unitary Ensemble (GUE) or Gaussian Orthogonal Ensemble (GOE)).
Before we get to these results, let us first discuss the simpler problem of studying the determinant of a random iid matrix , such as a real gaussian matrix (where all entries are independently and identically distributed using the standard real normal distribution ), a complex gaussian matrix (where all entries are independently and identically distributed using the standard complex normal distribution , thus the real and imaginary parts are independent with law ), or the random sign matrix (in which all entries are independently and identically distributed according to the Bernoulli distribution (with a chance of either sign). More generally, one can consider a matrix in which all the entries are independently and identically distributed with mean zero and variance .
We can expand using the Leibniz expansion
where ranges over the permutations of , and is the product
where ranges over the permutations of , and is the product
From the iid nature of the , we easily see that each has mean zero and variance one, and are pairwise uncorrelated as varies. We conclude that has mean zero and variance (an observation first made by Turán). In particular, from Chebyshev's inequality we see that is typically of size .
It turns out, though, that this is not quite best possible. This is easiest to explain in the real gaussian case, by performing a computation first made by Goodman. In this case, is clearly symmetrical, so we can focus attention on the magnitude . We can interpret this quantity geometrically as the volume of an -dimensional parallelopiped whose generating vectors are independent real gaussian vectors in (i.e. their coefficients are iid with law ). Using the classical base-times-height formula, we thus have
where is the -dimensional linear subspace of spanned by (note that , having an absolutely continuous joint distribution, are almost surely linearly independent). Taking logarithms, we conclude
where is the -dimensional linear subspace of spanned by (note that , having an absolutely continuous joint distribution, are almost surely linearly independent). Taking logarithms, we conclude
Now, we take advantage of a fundamental symmetry property of the Gaussian vector distribution, namely its invariance with respect to the orthogonal group . Because of this, we see that if we fix (and thus , the random variable has the same distribution as , or equivalently the distribution
where are iid copies of . As this distribution does not depend on the , we conclude that the law of is given by the sum of independent -variables:
A standard computation shows that each has mean and variance , and then a Taylor series (or Ito calculus) computation (using concentration of measure tools to control tails) shows that has mean and variance . As such, has mean and variance . Applying a suitable version of the central limit theorem, one obtains the asymptotic law
where denotes convergence in distribution. A bit more informally, we have
when is a real gaussian matrix; thus, for instance, the median value of is . At first glance, this appears to conflict with the second moment bound of Turán mentioned earlier, but once one recalls that has a second moment of , we see that the two facts are in fact perfectly consistent; the upper tail of the normal distribution in the exponent in (4) ends up dominating the second moment.
where denotes convergence in distribution. A bit more informally, we have
when is a real gaussian matrix; thus, for instance, the median value of is . At first glance, this appears to conflict with the second moment bound of Turán mentioned earlier, but once one recalls that has a second moment of , we see that the two facts are in fact perfectly consistent; the upper tail of the normal distribution in the exponent in (4) ends up dominating the second moment.
It turns out that the central limit theorem (3) is valid for any real iid matrix with mean zero, variance one, and an exponential decay condition on the entries; this was first claimed by Girko, though the arguments in that paper appear to be incomplete. Another proof of this result, with more quantitative bounds on the convergence rate has been recently obtained by Hoi Nguyen and Van Vu. The basic idea in these arguments is to express the sum in (2) in terms of a martingale and apply the martingale central limit theorem.
If one works with complex gaussian random matrices instead of real gaussian random matrices, the above computations change slightly (one has to replace the real distribution with the complex distribution, in which the are distributed according to the complex gaussian instead of the real one). At the end of the day, one ends up with the law
or more informally
(but note that this new asymptotic is still consistent with Turán's second moment calculation).
or more informally
(but note that this new asymptotic is still consistent with Turán's second moment calculation).
We can now turn to the results of our paper. Here, we replace the iid matrices by Wigner matrices , which are defined similarly but are constrained to be Hermitian (or real symmetric), thus for all . Model examples here include the Gaussian Unitary Ensemble (GUE), in which for and for , the Gaussian Orthogonal Ensemble (GOE), in which for and for , and the symmetric Bernoulli ensemble, in which for (with probability of either sign). In all cases, the upper triangular entries of the matrix are assumed to be jointly independent. For a more precise definition of the Wigner matrix ensembles we are considering, see the introduction to our paper.
The determinants of these matrices still have a Leibniz expansion. However, in the Wigner case, the mean and variance of the are slightly different, and what is worse, they are not all pairwise uncorrelated any more. For instance, the mean of is still usually zero, but equals in the exceptional case when is a perfect matching (i.e. the union of exactly -cycles, a possibility that can of course only happen when is even). As such, the mean still vanishes when is odd, but for even it is equal to
(the fraction here simply being the number of perfect matchings on vertices). Using Stirling's formula, one then computes that is comparable to when is large and even. The second moment calculation is more complicated (and uses facts about the distribution of cycles in random permutations, mentioned in this previous post), but one can compute that is comparable to for GUE and for GOE. (The discrepancy here comes from the fact that in the GOE case, and can correlate when contains reversals of -cycles of for , but this does not happen in the GUE case.) For GUE, much more precise asymptotics for the moments of the determinant are known, starting from the work of Brezin and Hikami, though we do not need these more sophisticated computations here.
Our main results are then as follows.
Theorem 1 Let be a Wigner matrix.
- If is drawn from GUE, then
- If is drawn from GOE, then
- The previous two results also hold for more general Wigner matrices, assuming that the real and imaginary parts are independent, a finite moment condition is satisfied, and the entries match moments with those of GOE or GUE to fourth order. (See the paper for a more precise formulation of the result.)
Thus, we informally have
when is drawn from GUE, or from another Wigner ensemble matching GUE to fourth order (and obeying some additional minor technical hypotheses); and
when is drawn from GOE, or from another Wigner ensemble matching GOE to fourth order. Again, these asymptotic limiting distributions are consistent with the asymptotic behaviour for the second moments.
The extension from the GUE or GOE case to more general Wigner ensembles is a fairly routine application of the four moment theorem for Wigner matrices, although for various technical reasons we do not quite use the existing four moment theorems in the literature, but adapt them to the log determinant. The main idea is to express the log-determinant as an integral
of the Stieltjes transform
of the Stieltjes transform
of . Strictly speaking, the integral in (7) is divergent at infinity (and also can be ill-behaved near zero), but this can be addressed by standard truncation and renormalisation arguments (combined with known facts about the least singular value of Wigner matrices), which we omit here. We then use a variant of the four moment theorem for the Stieltjes transform, as used by Erdos, Yau, and Yin (based on a previous four moment theorem for individual eigenvalues introduced by Van Vu and myself). The four moment theorem is proven by the now-standard Lindeberg exchange method, combined with the usual resolvent identities to control the behaviour of the resolvent (and hence the Stieltjes transform) with respect to modifying one or two entries, together with the delocalisation of eigenvector property (which in turn arises from local semicircle laws) to control the error terms.
Somewhat surprisingly (to us, at least), it turned out that it was the first part of the theorem (namely, the verification of the limiting law for the invariant ensembles GUE and GOE) that was more difficult than the extension to the Wigner case. Even in an ensemble as highly symmetric as GUE, the rows are no longer independent, and the formula (2) is basically useless for getting any non-trivial control on the log determinant. There is an explicit formula for the joint distribution of the eigenvalues of GUE (or GOE), which does eventually give the distribution of the cumulants of the log determinant, which then gives the required central limit theorem; but this is a lengthy computation, first performed by Delannay and Le Caer.
Following a suggestion of my colleague, Rowan Killip, we give an alternate proof of this central limit theorem in the GUE and GOE cases, by using a beautiful observation of Trotter, namely that the GUE or GOE ensemble can be conjugated into a tractable tridiagonal form. Let me state it just for GUE:
Proposition 2 (Tridiagonal form of GUE) Let be the random tridiagonal real symmetric matrix
where the are jointly independent real random variables, with being standard real Gaussians, and each having a -distribution:
where are iid complex gaussians. Let be drawn from GUE. Then the joint eigenvalue distribution of is identical to the joint eigenvalue distribution of .
Proof: Let be drawn from GUE. We can write
where is drawn from the GUE, , and is a random gaussian vector with all entries iid with distribution . Furthermore, are jointly independent.
We now apply the tridiagonal matrix algorithm. Let , then has the -distribution indicated in the proposition. We then conjugate by a unitary matrix that preserves the final basis vector , and maps to . Then we have
where is conjugate to . Now we make the crucial observation: because is distributed according to GUE (which is a unitarily invariant ensemble), and is a unitary matrix independent of , is also distributed according to GUE, and remains independent of both and .
We continue this process, expanding as
Applying a further unitary conjugation that fixes but maps to , we may replace by while transforming to another GUE matrix independent of . Iterating this process, we eventually obtain a coupling of to by unitary conjugations, and the claim follows.
The determinant of a tridiagonal matrix is not quite as simple as the determinant of a triangular matrix (in which it is simply the product of the diagonal entries), but it is pretty close: the determinant of the above matrix is given by solving the recursion
with and . Thus, instead of the product of a sequence of independent scalar distributions as in the gaussian matrix case, the determinant of GUE ends up being controlled by the product of a sequence of independent matrices whose entries are given by gaussians and distributions. In this case, one cannot immediately take logarithms and hope to get something for which the martingale central limit theorem can be applied, but some ad hoc manipulation of these matrix products eventually does make this strategy work. (Roughly speaking, one has to work with the logarithm of the Frobenius norm of the matrix first.)


text:
Marble Popper Games
Shoot marbles, match 3, and create chain reactions as you play free Marble Popper Games. Try before you buy!
#1 Hit Game
Jar of Marbles
Empty out the Jar of Marbles as quickly as you can in this fun and exciting Match 3 game!
All Genres
More Marble Popper Games
- 18
- Tonga
- 21
- Atlantis
- 22
- Zuma Deluxe

subset: commoncrawl


text:
Champions on top in free practice
As the second practice session came to a close today in preparation for Sunday's 6 Hours of Silverstone, the opening round of the FIA World Endurance Championship, it was Audi on top of the time sheets, starting the new season in fine style.
Although both sessions were declared wet, and brief showers were experienced during both, this afternoon's 90-minute practice was run in better conditions than earlier in the day. Lap times fell accordingly. André Lotterer set the fastest time of the day in the No.1 R18 Audi e-tron quattro which he shares with Benoît Tréluyer and Marcel Fässler, 1:43.217, but was only five hundredths of a second ahead of the No.2 Audi, Loîc Duval setting that car's best time of 1:43.263. The Frenchman is driving this weekend with the very experienced duo of Tom Kristensen and Allan McNish.


text:
Upcoming events
There are currently no local events scheduled.
In the meantime, you may find the following resources useful.
SAS Webinars
Choose a webinar on a topic that interests you.
SAS Training
Find a course or watch a tutorial on a wide range of topics.
Insights Center
Access the latest thinking on the subjects below, published and curated from experts around the world.


text:
Stork Report
Molly Sims is a Mom!
Us Weekly
It's a boy!
First-time parents Molly Sims and her husband Scott Stuber welcomed their son on Tuesday.
A rep confirms to Us Weekly the birth of baby Stuber and tells Us he weighed in a little over 7 lbs. No name has been revealed yet.
Sims, 39, married movie producer Stuber, 43, last September. The model/actress told Us last month she was beginning to get nervous about the birth of her son -- and so was Stuber!
"We did our first birthing class the other day," she said. "I swear to god I thought he was going to faint!"
But Sims did have her son's nursery ready.
"It's kind of bohemian, modern chic," she shared. "We have made it so cool. I'm so happy. It's my favorite room!"
Keep clicking to see photos of Molly throughout her pregnancy ...
Get out the vote
Sign up for celebrity breaking news!Please enter a valid email addressBy signing up I accept the Terms of Use and Privacy PolicyThanks for subscribing!Show CommentsNextWhen using Facebook Connect your image and name may display on Wonderwall. All Privacy Settings are controlled by Facebook
- button.AWW
- button.LOVE
- button.RAWR
- button.LOL
Like us on Facebook?
UP NEXTWeek in Photos
From Elle
From Crowd Ignite


text: Cuisinart - Pro Classic 7-Cup Food Processor - Brushed Chrome Model: DLC-10SBC Flag this deal What's the matter? Choose a reason Dead deal Duplicate Bad link Spam Inaccurate No value Alive again Edit deal cid681 Mar 18, 2013 I think it's finally time to buy one, I've been putting it off but this is just the size and brand I wanted. Reply 1 like chuckydealpl Mar 17, 2013 own one of these, this is an excellent quality processor, won't go wrong with cuisinart Reply 1 like glwrks Mar 17, 2013 This has the dough blade, which I would use somewhat. The warranty is superb, as are the reviews. It's a tried & true food processor at a good price. Reply Load More Comments


text:
recap 20 Mar 12
You should be able to deny access from your Firewall. I don't use Comodo so you will have to check on how to do this in section">http://personalfirewall.comodo.com/Comodo_Internet_Security_Pro_User_Guide.pdf&images=no">section 3.7
This thread is now locked and can not be replied to.
Should I upgrade to Windows 10? 8 reasons why you should upgrade to Windows 10... and 2 why you…
We are being sold the ability to spend money we don't have. And we love it
Microsoft claims its new VR lens system is better than Oculus Rift's
Apple Music preview: Why Spotify should be scared of Apple's impressive new music-streaming service


text:
Amazon-Wunschzettel von Henry Buckel
(Bücher, DVDs, weitere Produkte)
lastFM: Eliminate End of the School… - Last.fm
Dorothy invented the particular brilliant notion of magnetic locker wallpaper and shared with her father Henry Buckel. Luck for Debbie, the best ...
Search results for M Buckel - Pub360.com
Irondale, OH. Read more. 25. M Buckel Year of Birth: 1968. Thomas Henry Buckel · Steven J Buckel. Millington, MI. Read more. 26. M Buckel Year of Birth: 1958.
Henry Buckel - Inventor in Uncategorized - Canadian Patents
Henry Buckel is an inventor in Uncategorized. Search LandOfFree.com for Canadian inventors and patents. Rate them and share your experience with other ...
Telefonnummern, Adressen & Branchenkontakte
Page 11 - Indiana's Digital Historic Newspaper Program
Miss Marcella Buckel, of L-tperte. is spending the day with her pirrn;i, Mr. and Mrs . Henry Buckel. :i0 N. Main st. Join one of the largest fraternal ord rs in the world ...
Schlechte Nachrichten
109-09-0536 - Social Security Death Master File, free
Social Security number 109-09-0874 was issued to HENRY BUCKEL, who was born 07 April 1889 and, Death Master File says, died March 1967.
Henry Buckel Death Records
Find death records for Henry Buckel. Over 100 million deaths from around the world. Research your ancestors for free in our extensive genealogy databases.
findagrave: Alice Bittinger Buckel (1858 - 1937) - Find A Grave Memorial
+Charles Henry Buckel, b. September 02, 1885, Grantsville, MD, d. July 19, 1969, Garrett County, Md.. +Mary Jane Buckel, b. October 23, 1889 ...
Henry Buckel (1903 - 1984) | Sysoon memorial [en]
Henry Buckel is on Sysoon. Personal death notice and detailed information about the deceased person. [22783621-]
(1 - 4 von 26
Henry Buckel (1876 - 1946) - Genealogy - Geni
Genealogy for Henry Buckel (1876 - 1946) on Geni with over 140 million profiles of ancestors and living relatives. Page updated today.
Henry Buckel (deceased) - Genealogy - Geni
Genealogy for Henry Buckel (deceased) family tree on Geni, with over 140 million profiles of ancestors and living relatives.
Henry Buckel - Baltimore, Maryland (MD) | Crestleaf
View the Crestleaf profile about Henry Buckel. Get more information about Henry Buckel in Baltimore, MD.
Vera Buckel - Historical records and family trees - MyHeritage
Vera married Henry Buckel. They had 3 children: Alvin C Buckel and 2 other children. Vera lived in 1935, at address, Michigan. She lived in 1940, at address, ...
Tradiciones populares mexicanas - Google Books-Ergebnisseite
El Después de hacer sus pinitos, Pérez de Prián debutó como luchador profesional contra el estadounidense Henry Buckel en la plaza de toros de San Pablo.
Kopierbücher und Briefsammlung des Alexander von Siebold: ein...
... AokiShüzö AokiShüzö AokiShüzö Aoki Shüzö Aoki Shüzö [anon.] Aoki Shüzö [anon.] Beyl[?], von / Oberregierungsrat Bonar, Henry Buckel illeg. illeg. illeg.
Tradiciones populares mexicanas - Sonia Iglesias y Cabrera - Google...
"Un pueblo sin tradición es un pueblo sin porvenir", afirmaba el periodista y político colombiano Alberto Lleras con gran acierto. La esencia de las...
Longworth's American Almanac, New York Register, and City Directory...
James Rozel Charles, rigger rear 124 Bancker Rozés Justine, lace washer 115Ghamber Ruby Anthony, Hatter 7l Ann Ruck widow H rear 81 Henry Buckel ...
Full text of "The Registers of the Parish Church of Adel, in ...
Elisabeth, wife of Henry Buckel, of Arthington. 29. Robrt. Whitehead, of Kirsgill. yf ... 7. Elisabeth Whitehead, of Kirsgill. 15. Abraham Murgetrhode, of Cookridge.
Dokumente zum Namen
HENRY BUCKELL alias HENRY BUCKEL Born SALFORD ...
Partner websites are free to search but there may be a charge to view full transcriptions and download documents. Other services may also be available.
Lancashire OnLine Parish Clerk Project
Redvers Henry Buckel - [Child] of James Buckel & Mary Jane Born: 28 May 1900. Abode: 19 Eden Street Occupation: Iron Grinder Baptised by: R. Cecil Palmer
Artikel & Meinungen
RootsWeb: PAALLEGH-L [ALL] BUCKEL/BUCKELL/BUCKLE
I would greatly appreciate it: 1. THOMAS HENRY BUCKEL was born Abt. 1850 in England. He married ESTHER HILL. She was born Abt. 1850 ...
Los fallos de Darwin: Cosmovisión evolucionista - Parte 1 de 2
Henry Buckel, cuya teoría de la historia naturalista y evolucionista requería una análoga teoría de la ciencia, había ya indicado en el primer ...
RootsWeb: PAALLEGH-L [ALL] BUCKELL/BUCKEL/BUCKLE
THOMAS HENRY BUCKEL (BUCKELL) was born Abt. 1850 in England. He married. ESTHER HILL. She was born Abt. 1850 in England.
Privat - Sonstiges
Grave Records for Henry Buckel
Henry Buckel Grave Records. Locate grave and burial information for millions of veterans. Date of birth and death, burial and cemetery information, military ...
(1 - 20 von 28
HENRY BUCKEL (1889-1967) | Social Security Death Index ...
Henry Buckel was born in 1889 and passed away in 1967 . Access the Free SSDI Death Records on Mocavo and find your ancestors for free!
Kentucky Surnames - Kentucky Genealogy Index
Person Record: Henry Buckel - USA;KY 1910s. aka. Henry Buckel - KY, USA 2000s. Subscribe to Access. Copyright © 2013 - 2014 by Andrew J. Morris.
2013 Listings - Query List
Karen is the granddaughter of Vera Seeley Buckel, wife of Henry Buckel, and daughter of George and Isadore McFarren Seeley. Family papers indicate that ...
Buckel - LocateFamily.com
Buckel, Henry Click here to send a message to Henry Buckel (Male) ... Henry Buckel located in Louisville Kentucky United States of America ...
Edward Buckel - (162 records found) - Phone, Address ...
mylife. Edward Henry Buckel - Wink. mylife. Edward Henry Buckel - Wink. mylife. Edward James Buckel - Wink. mylife. Edward Henry Buckel - Wink.
F3194 - Blackburn
Buried, Wellsville Cem., Columbiana, OH Find all individuals with events at this location. Spouse, Thomas Henry Buckel | F3196. Married, 25 Jan 1942 ...
Looking for darvocet? | brandon darvocet - Tekcities.com
Henry Buckel E-mail: firstname.lastname@example.com. Bristol, CT, I think that they are likely nonlethal a good portion of the conjunctivitis program in ...
Matapeake Elementary School Lawrence Dunn 410-643-3105
Patrick Henry Buckel. Thomas Michael Bull. Shane DiAngelo Vincent. Megan Elizabeth Tozier. Zachary Paul Stump. Casey Patrick Schneck.
Ogemaw County Deaths 1985-1989 - MIGenWeb
Buckel, Harry Joseph, November 11, 1987, Henry Buckel, Susie Gildner. Budzinski, John A. December 22, 1985, Andrew Budzinski, Mary Soltysiak. Bugg, Ila ...
Show single schedule - Henley Census Group
Henry Buckel M 40 Not known, Same county. Mary Buckel F 35 Not known, Same county. Samuel Buckel M 7 Not known, Same county. Emma Buckel F 12 Not ...
The BUCKEL surname in the 1881 census | British Surnames
James Buckel Frederick Buckel Frank Buckel Edward Buckel Robert Buckel Henry Buckel Francis Buckel Walter Buckel Arthur Buckel Sidney Buckel A. Buckel
The Buckel Family History
Charles Buckel, 0, 2, 0, 2, 4, 2, 2, 2. James Buckel, 0, 1, 2, 3, 1, 2, 1, 5. Eliza Buckel, 0, 0, 2, 1, 3, 0, 2, 4. Henry Buckel, 0, 2, 1, 3, 4, 1, 1, 2. Thomas Buckel, 1, 1 , 0 ...
The Registers of the Parish Church of Adel, in the County of ...
Joseph Poppelwell, Henry Buckel, William Ingham. William Milthorp, maltster, and Mary Lamb, spinster, both of Adel. Licence. Witns.: Joseph Talbot, Wm. Bailey.
Upwey Marriages 1837-1921
Eliza, TRAVERS, 26, spin, Upwey, David TRAVERS, Labourer, David TRAVERS, Mary Richards STRAND. 142, 1872, Apr-09, Banns, Henry Buckel, SEYMOUR ...
Buckel - Bronx County - New York Genealogy
Andrew Buckel, born 1904. Andrew W Buckel, born 1930. Henry G Buckel, born 1890. Joan A Buckel, born 1939. Leah Buckel, born 1901. Rose Buckel, born ...
Henry Buckel b1872 - Altoona Blair Pennsylvania | 1930 United States...
1930 United States Census for Henry Buckel b1872 - Altoona Blair Pennsylvania is available for free on Mocavo. Start searching for your ancestors today.
Descendants of William Luce
Oregon Death Index, (Ancestry.com), "Electronic," Name: ??e, Henry Buckel County: Washington Death Date: 3 Apr 1929 Certificate: 92. 46.
Buckel - St Clair County
Euodias Buckel, born 1885, Clay. Henry Buckel, born 1912, Clay. Robert Buckel, born 1912, Clay. View the details and images from the 1940 Federal Census at ...
Buckel - Stark County
Fred Buckel, born 1892, Canton Township. Gladys M Buckel, born 1923, Canton Township. Henry Buckel, born 1897, Lexington Township. John Buckel, born ...
Craig Buckel
... Harry Buckel · Hartford Buckel · Hazel Buckel · Heather Buckel · Helen Buckel · Henrietta Buckel · Henry Buckel · Herb Buckel · Herbert Buckel · Hj Buckel ...
Verwandte Suchanfragen zu Henry Buckel
|Esther Hill
Thomas Henry
Robert Buckel
|Adam Henry
Person "Buckel" (2)
Vorname "Henry" (10907)
Name "Buckel" (288)
sortiert nach Relevanz / Datum


text:
About Stephanie
Stephanie Battista, Leadership Team and Senior Design Program Manager at Modern Edge. Stephanie oversees medical and wearable design programs at Modern Edge. For over a decade prior to joining Modern Edge, Stephanie was principal of a product design and development firm specializing in start-ups. Stephanie has expertise in a variety of industries including wearable technology, medical, textiles, and consumer goods. She brings design, project management, and sourcing expertise to Modern Edge. She is a Industrial Design graduate of Savannah College of Art and Design.
Areas of Expertise
Industrial Design (Consumer Electronics), Industrial Design Wearable Technology, Industrial Design Smart Health
An idea worth spreading
The future is ...wearable technology, in lifestyle, fitness, health, and medical it will infiltrate our daily lives and connect us with big business, data, and policy...it will facilitate communication in our daily lives far beyond what we can imagine...
I'm passionate about
Wearable Technology / Smart Health
Talk to me about
Design Strategy, Wearable Technology, + Smart Health...
People don't know I'm good at
Strategy, long term visionary...
My TED story
Ready to tell it...


text:
Read our Kuala Lumpur city guide for information about eating out, things to do and much more.View Kuala Lumpur Guide »
We're a boutique backpacker's hostel located in the heart of KL's historic district. We've lovingly converted a row... More Info »
We are a brand new hostel opened in June 2010, nestled right in the hub of the vibrant Kuala Lumpur city, More Info »
Selamat datang (welcome) to Travellers Palm Lodge, a friendly budget hostel located in the heart of Kuala Lumpur.... More Info »
Surrounded by towering skyscrapers and buildings in Kuala Lumpur, Reggae Mansion is a colonial style building which... More Info »
Central Lodge - The best located Guest House in Kuala Lumpur -200 meters from KL Sentral Station - the Transit Point... More Info »
A BACKPACKERS MECCA. A New Hostel In Kuala Lumpur. A Fresh and Modern Central Kuala Lumpur Accommodation. Great... More Info »
Located just a stone throw away from KL Sentral,a hub of public transportation. We offer inexpensive options to fit... More Info »
Paradiso is Kuala Lumpur\'s latest chic and totally cool Bed&Breakfast for the global backpackers. More Info »
HOT DEALS! Redeem 15% OFF stay at Fernloft Branch in Singapore and Melaka. Conditions applies, please email us... More Info »
Hotel Chinatown 2 offers comfortable accommodation and warm Chinese hospitality at affordable rates. We have a range... More Info »
Backpackers Travellers Inn offers no-frills, budget accommodation in a central location, close to Bukit Bintang and... More Info »
Orange Pekoe - formerly known as The Nest Guesthouse - is just 5 minutes away from the shopping district of Bukit... More Info »
Brand new guesthouse located centrally - in the heart of China Town. A home away from home style vibe greets you with... More Info »
Hotel ChinaTown Inn is a popular and established budget hotel ideally located in the heart of Kuala Lumpur's... More Info »
Classic Inn brings comfortable accommodation, friendly service and a homely atmosphere to Kuala Lumpur. Our private... More Info »
Daddy's Home is right next door to center market, only 200 meters walk from Pasar Seni LRT Station. China Town is 100... More Info »
MySpace Inn is located in Central Kuala Lumpur, an easy 3 minute walk from Titiwangsa Monorail/LRT and the Pekeliling... More Info »
The Longhouse is a brand new bed & breakfast guesthouse that will make your stay pleasant with its cozy interior,... More Info »
Every room equips with AIR-CONDITIONER, PRIVATE TOILET & BATHROOM, REFRIGERATOR, TV, WARDROBE, TOWELS AND TOILETRIES... More Info »
Renovated in 2012, the City Campus Lodge & Hotel guarantees students and guests a pleasant stay, whether in Kuala... More Info »
Video guides, travel podcasts, city maps & free city guides. What to eat, where to stay and things to do and see - our travel guides have it all.Find a guide »
There's nothing like having your own transport to get around and when travelling the world it's no different.Read more »
We offer the most comprehensive selection of hostels on the internet with over 35,000 great places to stay in 180 countries. The ultimate online resource for hostels everywhere.Find great value hostels »


text:
Stores »
Simply sweet notecards for any occasion. Love these!
We are a community of shoppers sharing our best finds from across the web.
© 2015 ThisNext, Inc. ThisNext is a member of BUZZMEDIA Entertainment, a division of BUZZMEDIA.
Create your own free digital magazine at our new site Glossi.com