open access

https://hdl.handle.net/2115/85983

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

英語

lace expansion

2020-04

Research Institute for Mathematical Sciences, Kyoto University

RIMS Kokyuroku Bessatsu

B79

51-62

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.

学術雑誌論文

Version of Record

1881-6193

AA12196120