HIGH-DIMENSIONAL NEAR-CRITICAL PERCOLATION AND THE TORUS PLATEAU

• 成果类型: Article
• 署名作者: Hutchroft, Tom; Michta, Emmanue; Slade, Gordon
• 署名单位: California Institute of Technology; University of British Columbia
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1608
• 发表日期: 2023
• 页码: 580-625
• 关键词: critical 2-point functions; field critical-behavior; critical exponents; random subgraphs; finite graphs; lace expansion; long-range; inequalities; TREE

摘要

We consider percolation on Z(d) and on the d-dimensional discrete torus, in dimensions d >= 11 for the nearest-neighbour model and in dimensions d > 6 for spread-out models. For Z(d) we employ a wide range of techniques and previous results to prove that there exist positive constants c and C such that the slightly subcritical two-point function and one-arm probabilities satisfy Ppc-epsilon(0 <-> x) <= C/||x||(d-2) e(-c epsilon 1/2) ||x|| c/r(2)e -C epsilon(1/2r) <= Ppc-epsilon (0 <-> partial derivative [-r,r](d)) <= C/r(2) e(-c epsilon 1/2)r . Using this, we prove that throughout the critical window the torus two-point function has a plateau, meaning that it decays for small x as ||x||(-(d-2)) but for large x is essentially constant and of order V (-2/3) where V is the volume of the torus. The plateau for the two-point function leads immediately to a proof of the torus triangle condition, which is known to have many implica-tions for the critical behaviour on the torus, and also leads to a proof that the critical values on the torus and on Z(d) are separated by a multiple of V-1/3. The torus triangle condition and the size of the separation of critical points have been proved previously, but our proofs are different and are direct con-sequences of the bound on the Z(d) two-point function. In particular, we use results derived from the lace expansion on Z(d), but in contrast to previous work on high-dimensional torus percolation, we do not need or use a separate torus lace expansion.