Crossover phenomena in the critical behavior for long-range models with power-law couplings (Stochastic Analysis on Large Scale Interacting Systems)

Abstract

This is a short review of the two papers [9, 10] on the x-space asymptotics of the critical two-point function Gpc (x) for the long-range models of self-avoiding walk, percolation and the Ising model on Zd, defined by the translation-invariant power-law step-distribution/coupling D(x)|x|-d-αor some α > 0. Let S1(x) be the random-walk Green function generated by D. We have shown that S1(x) changes its asymptotic behavior from Newton (α > 2) to Riesz (α < 2), with log correction at α = 2; Gpc(x) - A/pc S1(x) as|x| → ∞ in dimensions higher than (or equal to, if α = 2) the upper critical dimension dc (with sufficiently large spread-out parameter L). The model-dependent A and dc exhibit crossover at α = 2. The keys to the proof are (i) detailed analysis on the underlying random walk to derive sharp asymptotics of S1, (ii) bounds on convolutions of power functions (with log corrections, if α = 2) to optimally control the lace-expansion coefficients π(n)p, and (iii) probabilistic interpretation (valid only when α ≤ 2) of the convolution of D and a function Πp of the alternating series Σ∞n=0(-1)nπ(n)p. We outline the proof, emphasizing the above key elements for percolation in particular.