Spread-out limit of the critical points for lattice trees and lattice animals in dimensions d > 8
COMBINATORICS PROBABILITY & COMPUTING（2023）
摘要
A spread-out lattice animal is a finite connected set of edges in {{x,y}subset of Z(d):0= 1. In this paper, we show that pc=1/e+CL-d+O(L-d-1) for all d>8, where the model-dependent constant C has the random-walk representation C-LT = (infinity)& sum;(n+1)(n=2 )/U-2e(& lowast;n)(o), C-LA = C-LT-(1)/(2e)2 (infinity)& sum;U-n=3(& lowast;n)(o),where U(& lowast;n )is the n-fold convolution of the uniform distribution on the d-dimensional ball {x is an element of R-d:parallel to x parallel to <= 1}. The proof is based on a novel use of the lace expansion for the 2-point function and detailed analysis of the 1-point function at a certain value of p that is designed to make the analysis extremely simple.
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关键词
Lattice trees,lattice animals,critical phenomena
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