Crossover phenomena in the critical behavior for long-range models with power-law couplings
Akira Sakai

Flocking of Active Particles in a Turbulent Flow
    Samriddhi Sankar Ray
    December 30, 2023
================================================






This is a short review of the two papers <cit.> on the x-space 
asymptotics of the critical two-point function G_(x) for the long-range 
models of self-avoiding walk, percolation and the Ising model on , defined 
by the translation-invariant power-law step-distribution/coupling 
D(x)∝|x|^-d-α for some α>0.  Let S_1(x) be the 
random-walk Green function generated by D.  We have shown that


  * S_1(x) changes its asymptotic behavior from Newton (α>2) to Riesz 
(α<2), with log correction at α=2;

  * G_(x)∼A/S_1(x) as |x|→∞ in dimensions higher than 
(or equal to, if α=2) the upper critical dimension  (with 
sufficiently large spread-out parameter L).  The model-dependent A and 
 exhibit crossover at α=2.  

The keys to the proof are (i) detailed analysis on the underlying random walk 
to derive sharp asymptotics of S_1, (ii) bounds on convolutions of power 
functions (with log corrections, if α=2) to optimally control the 
lace-expansion coefficients π_p^(n), and (iii) probabilistic 
interpretation (valid only when α≤2) of the convolution of D and 
a function _p of the alternating series 
∑_n=0^∞(-1)^nπ_p^(n).  We outline the proof, emphasizing 
the above key elements for percolation in particular.






§ INTRODUCTION AND THE MAIN RESULTS

Since the dawn of research on phase transitions and critical behavior, 
it has been standard to investigate short-range models, among which 
the nearest-neighbor model on  is the most popular.  Thanks to 
intensive studies for more than half a century, nearest-neighbor bond 
percolation is now known to exhibit a phase transition for all d≥2 
and mean-field behavior (i.e., the critical two-point function G_(x) 
decays as |x|^2--d with the mean-field value =0) for all 
d≥11 <cit.>.  Believing in universality, we expect the 
mean-field behavior for all dimensions above the upper-critical dimension 
=6 for short-range percolation <cit.>.

Recently, long-range random walk and statistical-mechanical models defined 
by the power-law step-distribution/coupling D(x)∝|x|^-d-α, 
α>0, have regained popularity, due to unconventional macroscopic 
behavior <cit.>.  Among those 
references, an infrared bound and mean-field behavior are proven for 
long-range oriented percolation (OP for short) on ^d>×_+ 
<cit.> and for long-range models of percolation, self-avoiding walk 
(SAW for short) and the Ising model on ^d> <cit.>, where

    dcdef=(α∧2)×
     2     [OP, SAW & Ising],
    
     3   [percolation].

Also, an asymptotic expression of the gyration radius for long-range models of 
SAW and OP for d> are proven in <cit.>.  In physics, Brezin, Parisi 
and Ricci-Tersenghi <cit.> conjectured that G_(x) would decay as 
|x|^α∧(2-)-d if α2-, and as 
|x|^α-d/log|x| if α=2-.  We have shown in <cit.> 
that the conjectured behavior holds true for d> (= with 
α=∞), because =0, with sufficiently large spread-out 
parameter L <cit.>.  In fact, the obtained results are much 
stronger, as summarized as follows.



Let α>0, L≥1 and 
D(x)≍1L^d(|x|/L∨ 1)^-d-α, i.e., 

    Ddef^∃ c>0, ^∀ x∈, ^∀ L∈[1,∞): c≤D(x)/1/L^d(|x|/L∨ 1)^-d-α≤1/c.

Let

    gammaγ_α=Γ(d-α∧2/2)/2^α∧2π^d/2Γ(α∧2/2),            
    v_α=
     lim_|k|→01-D̂(k)/|k|^α∧2   [α2],
    lim_|k|→01-D̂(k)/|k|^2log(1/|k|)     [α=2],

where D̂(k)=∑_x∈e^ik· xD(x).
Then, for all d>α∧2, the random-walk Green function S_1(x) 
generated by the step distribution D exhibits the following asymptotic 
behavior: there is an ϵ>0 such that, as |x|→∞,

    S1asy
    S_1(x)=γ_α/v_α/|x|^d-α∧2×(1+O(L^ϵ)/|x|^ϵ)   [α2],
    1/log|x|(1+O(1)/(log|x|)^ϵ)     [α=2],

where the O(1) term is independent of L.




Let D be the same as in Theorem <ref> and recall the definition 
dcdef of .  Suppose d> for α2 or that d≥ 
for α=2.  For α>2, we also assume a bound on the “derivative" 
of D (see the last part of Section <ref>).  Then, there is an 
L_0(d)<∞ such that, for any L≥ L_0, there are 
A=1+O(L^-2)α>2 and ϵ>0 such that, as |x|→∞,

    mainasy
    G_(x)=A/γ_α/v_α/|x|^d-α∧2×(1+O(L^ϵ)/|x|^ϵ)   [α2],
    1/log|x|(1+O(1)/(log|x|)^ϵ)     [α=2],

where the O(1) term is independent of L.


In short, the critical two-point function G_(x) exhibits the same 
asymptotic behavior as S_1(x), modulo multiplication of the model-dependent 
constant A/, for all d> (with large spread-out parameter L) and, 
most interestingly, for d= when α=2.  For d∈(,), which is 
not empty for α<2 and in which  is believed to be nonzero, 
Theorem <ref> claims that G_(x) decays as |x|^α-d, 
not as |x|^2--d.  This power-law behavior has been 
extended even below  by Lohmann, Slade and Wallace <cit.> using 
a rigorous version of the ε-expansion.  





§ KEY IDEAS FOR THE PROOF OF THEOREM <REF>

Let D^*n be the n-fold convolution of D (i.e., the n-step 
distribution) and denote by S_q the random-walk Green function generated 
by D with survival rate q∈[0,1]:

    D^*n(x)=(D^*(n-1)*D)(x)≡∑_yD^*(n-1)(y) D(x-y),
    
    S_q(x)=∑_n=0^∞ q^nD^*n(x).

Let

    veeex_r=π/2(|x|∨ r)    [x∈, 1≤ r<∞],

where |·| is the Euclidean norm.  Suppose that, as explained in 
Ddef, D(x) decays as 

    Dredef
    D(x)≍ L^-dxL_1^-d-α≡ L^αx_L^-d-α.

An example of D is the following compound zeta distribution <cit.>:

    comp-zeta
    D(x)=∑_t∈U_L^*t(x) t^-1-α/2/ζ(1+α/2)    [x∈],

where U_L is the uniform distribution over the d-dimensional box 
of side-length 2L.

The step distribution D in Dredef satisfies the following properties 
(D1)–(D3) that are essential to the proof of S1asy.


  (D1)
k-space bounds <cit.> (and <cit.>): 
^∃Δ=Δ(L)∈(0,1) such that

    1-hatDbd1
    1-D̂(k)
     
     <2-Δ     [^∀ k∈[-π,π]^d],
    
     >Δ   [|k|>1/L],

and for |k|≤1/L,

    1-hatDbd2
    1-D̂(k)≍ (L|k|)^α∧2×
     1   [α2],
    logπ/2L|k|     [α=2].


  (D2)
k-space asymptotics <cit.> (and <cit.>): 
^∃ϵ>0 such that, as |k|→0,

    1-hatDasy
    1-D̂(k)=v_α|k|^α∧2×(1+O(L^ϵ|k|^ϵ))   [α2],
    (log1/L|k|+O(1))     [α=2],

where the constant in the O(1) term is independent of L.

  (D3)
x-space bounds <cit.> (and <cit.>): 
^∀ n∈ and ^∀ x∈,

    D^*n_∞   ≤ O(L^-d)×
     n^-d/(α∧2)   [α2],
    
     (nlogπ n/2)^-d/2     [α=2],
     Dnsupbd
    
    D^*n(x)   ≤ nO(L^α∧2)/x_L^d+α∧2×
     1   [α2],
    logx/L_1     [α=2].
     Dnxbd



For example, to show 1-hatDbd2 for |k|≤1/L, we first split the sum 
as 

    1-D̂(k)≍ L^α∑_xx_L^-d-α(1-cos k·
     x) (|x|<L+L≤|x|≤π/2|k|+|x|>π/2|k|).

It is easy to see that the contributions from the first and third indicators 
are O(L^2|k|^2) and O(L^α|k|^α), respectively.  The contribution 
from the second indicator is the main term since

    L^α∑_L≤|x|≤π/2|k|x_L^-d-α(1-cos k· x)
       ≍ L^α|k|^2∑_L≤|x|≤π/2|k||x|^-d-α+2
       ≍
     (L|k|)^α∧2   [α2],
    
     (L|k|)^2logπ/2L|k|   [α=2].


To prove S1asy, we first rewrite S_1(x) for the transient case 
d>α∧2 as

    S_1(x)=∫_[-π,π]^dd^dk/(2π)^d e^-ik· x/1-D̂(k)   =∫_0^∞dt∫_[-π,π]^dd^dk/(2π)^d e^-ik· x-t(1-D̂(k))
       =∫_0^∞dt∫_|k|≤ Rd^dk/(2π)^d e^-ik
     · x-t(1-D̂(k))+E_1,

where R is arbitrary for the moment.  Then, by replacing 1-D̂(k) by 
its limit 1-hatDasy, we can further rewrite S_1(x) for α2 as

    S_1(x)=∫_0^∞dt∫_d^dk/(2π)^d e^-ik·
     x-v_α t|k|^α∧2+E_1+E_2,

and for α=2 as

    S_1(x)=∫_0^∞dt∫_d^dk/(2π)^d e^-ik·
     x-v_2t|k|^2log1/L|k|+E_1+E_2.

Since

    ∫_0^∞dt e^-v_α t|k|^α∧2=1/v_α
     |k|^α∧2=1/v_αΓ(α∧2/2)∫_0^∞dt/t t^(α∧2)/2e^-t|k|^2,

we readily obtain for α2 that

    S_1(x)-E_1-E_2=1/v_αΓ(α∧2/2)∫_0^∞dt/t t^(α∧2)/2∫_d^d
     k/(2π)^d e^-ik· x-t|k|^2_= (4π t)^-d/2exp(-|x|^2/(4t))
     =γ_α/v_α/|x|^d-α∧2.

Using the k-space and x-space bounds (D1) and (D3) and choosing R 
accordingly (as in <cit.>), we can show that E_1+E_2 is the 
error term in S1asy.  See <cit.> for more details.  

For α=2, we change variables as ξ=x/|x|, κ=|x|k and 
τ=v_2t/|x|^2log|x|/L to obtain

    S_1(x)-E_1-E_2   =|x|^-d∫_0^∞dt∫_d^dκ/(2π)^d exp(-iκ·ξ-v_2t|κ|^2/|x|^2log|x|/L|κ|)
       =|x|^2-d/v_2log|x|/L∫_0^∞dτ∫_d^dκ/(2π)^d exp(-iκ·ξ-τ|κ|^2log|x|/L|κ|/log|x|/L)
       =|x|^2-d/v_2log|x|/L∫_0^∞dτ∫_d^dκ/(2π)^d e^-iκ·ξ-τ
     |κ|^2_= γ_2+E_3.

Again, by using the k-space and x-space bounds on D and choosing R 
accordingly (as in <cit.>), we can show that E_1+E_2+E_3 is the 
error term in S1asy.  See <cit.> for more details.  
This completes the sketch proof of Theorem <ref>.





§ KEY IDEAS FOR THE PROOF OF THEOREM <REF>

The proof of Theorem <ref> is based on the lace expansion, which is 
one of the few methods to prove mean-field results mathematically rigorously.  
Since its invention by Brydges and Spencer for weakly SAW <cit.>, 
the method has been extended to strictly SAW <cit.>, oriented/unoriented 
percolation <cit.>, lattice trees and lattice animals <cit.>, 
the contact process <cit.>, the Ising and φ^4 models <cit.>.

The lace expansion yields a formal recursion equation for the two-point 
function G_p(x), which is similar to the recursion equation for the 
random-walk Green function S_p(x).  For (strictly) SAW, G_p(x) is defined as

    SAW-2pt
    G_p(x)=∑_ω:o→ xp^|ω|∏_j=1^|ω|
     D(ω_j-ω_j-1)∏_s<t(1-δ_ω_s,ω_t),

where the sum is over the paths ω from o to x.  The contribution 
from the zero-step walk is regarded as δ_o,x.  The last product over 
s,t is either 0 or 1 depending on whether or not ω intersects to 
itself.  

For Bernoulli bond percolation, in which each bond {u,v} is occupied with 
probability pD(v-u) independently of the other bonds, the two-point function 
is defined as 

    G_p(x)=_p(o x),

where _p is the induced law from the above bond-occupation probability 
(p(1-D(o)) is the expected number of occupied bonds per vertex), and 
{o x} is the event that either x=o or there is a self-avoiding path 
of occupied bonds from o to x.

For the Ising model, see, e.g., <cit.>.

Due to monotonicity in p and subadditivity in self-avoiding paths, the 
critical point  is characterized by the divergence of the susceptibility 
χ_p for all models, as follows:

    χ_p=∑_xG_p(x),      =sup{p≥0:χ_p<∞}.


The proof of Theorem <ref> consists of the following two steps:


  Step 1:
Prove that G_p(x) is bounded by 
2λx_L^α∧2-d if α2 and by 
2λx_L^2-d/logx/L_1 if α=2, uniformly in 
x∈ and p<, where

    lambdaλ=
     sup_x oS_1(x)x_L^d-α∧2   [α2],
    sup_x oS_1(x)x_L^d-2logxL_1     [α=2],

which is of order L^-α∧2, by Theorem <ref>.  

  Step 2:
Use the lace expansion as a recursion equation for G_(x) to derive 
its asymptotic expression.


To complete Step 2 is rather straightforward as soon as 
Step 1 is completed; see <cit.> for α2 
and <cit.> for α=2.  To complete Step 1, 
it suffices to show that g_p, defined as

    g-def
    g_p=
      p∨sup_x oG_p(x)/λx_L^α∧2-d   [α2],
     p∨sup_x oG_p(x)/λx_L^2-d/logx/L_1   [α=2],

satisfies the following three properties:


  * g_1≤1.

  (S1.2)
g_p is continuous (and nondecreasing) in p∈[1,).

  (S1.3)
g_p≤3 implies g_p≤2 for every p∈(1,), if λ≪1.

The third property implies that there is a prohibited region in the 
p–g_p plane.  Therefore, g_p is either ≤2 or >3, as long as 
p∈(1,).  However, due to the continuity (S1.2) with the initial 
condition (S1.1), the possibility of g_p>3 is eliminated.  This completes 
Step 1.

(S1.1)–(S1.2) are not so difficult, due to 
<cit.>.  To show (S1.3), 
we use the lace expansion, which is formally written as

    lace-intro
    G_p(x)=_p(x)+(_p*pD*G_p)(x),

where (cf., <cit.>)

    vPi-def_p(x)=
     δ_o,x+∑_n=1^∞(-pD(o)δ+π_p)^*n(x)
         [SAW],
    π_p(x)+∑_n=1^∞(-pD(o))^nπ_p^*(n+1)(x)     [Ising & percolation].

Here, π_p is the alternating series of the nonnegative lace-expansion 
coefficients {π_p^(n)}_n=0^∞ (π_p^(0)≡0 for 
SAW):

    exp-coeffπ_p(x)=∑_n=0^n(-1)^nπ_p^(n)(x).


The proof of Item (S1.3) goes as follows.  


  * Bound π_p^(n) in terms of G_p by using correlation inequalities, 
such as the BK inequality for percolation <cit.>.

  * Derive an optimal x-space bound on _p in vPi-def by applying 
the hypothesis g_p≤3 to the bounds on π_p^(n) obtained in (i) and 
using convolution bounds (see below) on power functions, with log corrections 
for α=2.

  * Prove the improved bound g_p≤2 by applying the bound on _p 
obtained in (ii) to lace-intro.


From now on, we restrict our attention to percolation.  By the BK inequality, 
the first few terms are bounded as 

    π_p^(0)(x)≤ G_p(x)^2,            π_p^(1)(x)≤ o -7pt
        < g r a p h i c s >
     x,            π_p^(2)(x)≤ o -9pt
        < g r a p h i c s >
     x 
     +⋯,

where each line segment represents G_p, small filled rectangles are pD and 
unlabeled vertices are summed over .  For more explanation on those 
diagrammatic expressions, we refer to the original paper <cit.>.  Then, 
we use g_p≤3 and the following convolution bounds:

[Lemma 3.5 of <cit.>]
For a_1≥ b_1>0 with a_1+b_1≥ d, and for a_2,b_2≥0 with
a_2≥ b_2 when a_1=b_1, there is an L-independent constant
C=C(d,a_1,a_2,b_1,b_2)<∞ such that

    conv1   ∑_y∈x-y_L^-a_1/(logx-y/L_1)^a_2y_L^-b_1/(logy/L_1)^b_2
       ≤C x_L^-b_1/(logx/L_1)^b_2×
     L^d-a_1   [a_1>d],
    loglogx/L_1   [a_1=d, a_2=1],
    
     (logx/L_1)^0∨(1-a_2)   [a_1=d, a_21],
    x_L^d-a_1   [a_1<d, a_1+b_1>d],
    x_L^b_1(logx/L_1)^0∨(1-a_2)   [a_1<d, a_1+b_1=d,
       a_2+b_2>1].



Take π_p^(1)(x) for α=2, for example.   By repeated 
applications of the above convolution bounds, we can reduce the number of 
vertices (and line segments) one by one, as depicted as follows:

    conv-appl
    o 
     -7pt
        < g r a p h i c s >
     x g_p≤3
    
     d≥4≲ o -7pt
        < g r a p h i c s >
     x +
      o -7pt
        < g r a p h i c s >
     x g_p≤3
     d≥6≲ o -7pt
        < g r a p h i c s >
     x.



Let v be the unlabeled top-right vertex in the leftmost figure at which three 
line segments (each in red, blue and black) meet, and let y,z be the other 
end vertices of the horizontal (in red) and vertical (in black) line segments, 
respectively.  In the first inequality, we use conv1 between the 
vertical line segment and one of the other two line segments, depending on 
whether |x-v|≥|y-v| or |x-v|≤|y-v|.  If |x-v|≤|y-v|, then 
|x-y|≤|x-v|+|y-v|≤2|y-v| and therefore

    ∑_v:|x-v|≤|y-v|x-v_L^2-d/logx-v/L_1y-v_L^2-d/logy-v/L_1z-v_L^2-d/logz-v/L_1
    ≤x-y/2_L^2-d/logx-y/2L_1∑_vx-v_L^2-d/logx-v/L_1z-v_L^2-d/logz-v/L_1   d≥4≤^∃ C'x-y_L^2-d/logx-y/L_1x-z_L^4-d/logx-z/L_1_blue-dotted,

which is depicted as the left figure in the middle expression in 
conv-appl.  Then, by gathering all line segments meeting at z 
(denote the other end vertex of the horizontal line segment by u) 
and using conv1 again, we obtain

    ∑_zx-z_L^(4-d)+(2-d)/(logx-z/L_1)^2u-z_L^2-d/logu-z/L_1d≥6≤
     Cx-u_L^2-d/logx-u/L_1,

which yields the rightmost figure of conv-appl.  We should emphasize 
that the above bound holds even at =6, because of the log-squared 
term in the denominator.  This is one of the reasons why the mean-field 
results[The bubble condition G_^*2(o)<∞ for SAW/the 
Ising model and the triangle condition G_^*3(o)<∞ for percolation 
are sufficient conditions for the susceptibility χ_p and other observables 
to exhibit their mean-field behavior.  The log correction for α=2 
is the key to extend the mean-field results down to d= since, for 
example, the tail of the sum in the triangle condition can be estimated, 
for any R>1, as 

    ∑_x:|x|>RG_(x) G_^*2(x)d≥4≲∫_R^∞dr/r r^6-d/(log r)^2d
     ≥6<∞.

] 
hold for d≥ (including equality) when α=2.

The other case |x-v|≥|y-v| can be evaluated similarly, and we refrain 
from showing it here.


Applying the same analysis to the other π_p^(n) and using 
vPi-def–exp-coeff, we can get (cf., <cit.> and <cit.>)

    varPi-bd
    |_p(x)-δ_o,x|   ≤ O(L^-d)δ_o,x+O(λ^2)×x_L^(α∧2-d)ℓ   [α2],
    
     (x_L^2-d/logx/L_1)^ℓ     [α=2],

where 

    elldefℓ=
     
     2   [percolation],
    
     3     [SAW & Ising].

Notice from varPi-bd that, if α<2 and d> or if α=2 
and d≥, then _p*D in lace-intro can be treated, after 
normalization, as a probability distribution.  For α=2, for example, 
there are finite constants c,c',c” such that

    +ity
    (_p*D)(x)   varPi-bd≥   (1-cL^-d)D(x)-c'λ^2
     ∑_yy_L^ℓ(2-d)/(logy/L_1)^ℓD(x-y)
       Lemma <ref>≥   (1-cL^-d-c”λ^3)D(x),

which is positive for all x, if λ≪1.  Therefore, 

    new1step(x)=(_p*D)(x)/_p(0)

is a probability distribution that satisfies all the properties in (D1)–(D3), 
and its Green function ∑_n=0^∞^*n(x) is bounded by 
(1+O(λ^3))S_1(x) for every x (see <cit.> for more 
details).  By varPi-bd and Lemma <ref>, we obtain 
that, for x o,

    G_p(x)≤(1+O(λ^3))(_p*S_1)(x)
       ≤(1+O(λ^3))S_1(x)+O(λ^4)x_L^2-d/logx/L_1
       λ≪1≤2λx_L^2-d/logx/L_1,

as required.  This completes all the steps (i)–(iii) for α≤2.

If α>2, then we can no longer interpret _p*D as a probability 
distribution, because the second term in +ity decays slower than D; 
this is why the model-dependent multiplicative constant A in mainasy 
is reduced to 1 only when α≤2.  To overcome this difficulty for 
α>2, we assume that the “derivative" of the n-step distribution 
D^*n obeys the following bound: for |y|≤1/3|x|,

    Dnxdiffbd|D^*n(x)-D^*n(x+y)+D^*n(x-y)/2|≤ n O(L^α∧2) y_L^2/x_L^d+α∧2+2.

We have shown in <cit.> that the compound zeta distribution 
comp-zeta for α2 satisfies the above assumption.  
See <cit.> for more details.





§ ACKNOWLEDGEMENTS

This work was supported in part by JSPS KAKENHI Grant Number 18K03406, 
and in part by JST CREST Grant Number JP22180021.  I would like to thank 
Lung-Chi Chen for the long-time collaboration <cit.> on 
the long-range models defined by the power-law decaying 
step-distribution/coupling.  I would also like to thank the organizers of 
17^th International Symposium “Stochastic Analysis on Large-scale 
Interacting Systems" at RIMS, Kyoto University, during November 5–8, 2018, 
for the opportunity to speak on the topic of this article.  Finally, I would 
like to express my gratitude to the referee for positive feedback and valuable 
suggestions to improve presentation of this paper.



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csIIChen L.-C. and Sakai A.,
Critical behavior and the limit distribution for long-range oriented
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Probab. Theory Relat. Fields, 145 (2009), 435–458.

csIIIChen L.-C. and Sakai A.,
Asymptotic behavior of the gyration radius for long-range
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csIVChen L.-C. and Sakai A.,
Critical two-point functions for long-range statistical-mechanical
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csVChen L.-C. and Sakai A.,
Critical two-point functions for long-range statistical-mechanical
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fh17aFitzner R. and Hofstad R.v.d.,
Generalized approach to the non-backtracking lace expansion,
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fh17bFitzner R. and Hofstad R.v.d.,
Mean-field behavior for nearest-neighbor percolation in d>10,
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hhs03Hara T., Hofstad R.v.d. and Slade G.,
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hs90pHara T. and Slade G.,
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hs90lHara T. and Slade G.,
On the upper critical dimension of lattice trees and lattice animals,
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hs92Hara T. and Slade G.,
Self-avoiding walk in five or more dimensions. I.
The critical behaviour,
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hhs08Heydenreich M., Hofstad R.v.d. and Sakai A.,
Mean-field behavior for long- and finite-range Ising model,
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J. Stat. Phys., 132 (2008), 1001–1049.

lsw17Lohmann M., Slade G. and Wallace B.C.,
Critical two-point function for long-range O(n) models below 
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ms93Madras N. and Slade G.,
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ny93Nguyen B.G. and Yang W-S.,
Triangle condition for oriented percolation in high dimensions,
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s01Sakai A.,
Mean-field critical behavior for the contact process,
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s07Sakai A.,
Lace expansion for the Ising model,
Commun. Math. Phys., 272 (2007), 283–344.

s15Sakai A.,
Application of the lace expansion to the φ^4 model, 
Commun. Math. Phys., 336 (2015), 619–648.


