Sharpness of the phase transition in percolation models
type: Item
instance of: scholarly article
title: Sharpness of the phase transition in percolation models
zbMATH Open document ID: 0618.60098
DOI: 10.1007/BF01212322
author: David J. Barsky
author: Michael Aizenman
published in: Communications in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: The equality of two critical points -- the percolation threshold \(p_ H\) and the point \(p_ T\) where the cluster size distribution ceases to decay exponentially -- is proven for all translation invariant independent percolation models on homogeneous \(d\)-dimensional lattices (\(d\geq 1\)). The analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter \(M(\beta,h)\), which for \(h=0\) reduces to the percolation density \(P_{\infty}\) -- at the bond density \(p=1- e^{-\beta}\) in the single parameter case. These are: (1) \(M\leq h\partial M/\partial h+M^ 2+\beta M\partial M/\partial \beta\), and (2) \(\partial M/\partial \beta \leq | J| M\partial M/\partial h\). Inequality (1) is intriguing in that its derivation provides yet another hint of a ''\(\phi^ 3\) structure'' in percolation models. Moreover, through the elimination of one of its derivatives, (1) yields a pair of ordinary differential inequalities which provide information on the critical exponents \({\hat \beta}\) and \(\delta\). One of these resembles an Ising model inequality of Fröhlich and Sokal and yields the mean field bound \(\delta\geq 2\), and the other implies the result of Chayes and Chayes that \({\hat \beta}\leq 1\). An inequality identical to (2) is known for Ising models, where it provides the basis for Newman's universal relation \(\hat\beta(\delta-1)\geq 1\) and for certain extrapolation principles, which are now made applicable also to independent percolation. These results apply to both finite and long range models, with or without orientation, and extend to periodic and weakly inhomogeneous systems.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B43
zbMATH DE Number: 4003223
zbMATH Keywords: phase transition
zbMATH Keywords: percolation threshold
zbMATH Keywords: partial differential inequalities
zbMATH Keywords: percolation density
zbMATH Keywords: Ising models
Wikidata QID: Q56144767
MaRDI profile type: Publication
cites work: The critical probability of bond percolation on the square lattice equals 1/2
cites work: On the critical percolation probabilities
cites work: Discontinuity of the percolation density in one dimensional \(1/| x- y| ^ 2\) percolation models
cites work: Discontinuity of the magnetization in one-dimensional \(1/| x-y| ^ 2\) Ising and Potts models.
cites work: Q3678367
cites work: Critical points and intermediate phases on wedges of Zd
cites work: Percolation Processes: Lower Bounds for the Critical Probability
cites work: Tree graph inequalities and critical behavior in percolation models
cites work: One dimensional \(1/| j-i| ^ s\) percolation models: The existence of a transition for s\(\leq 2\)
cites work: On the critical behavior of the magnetization in high-dimensional Ising models
cites work: Q3810662
cites work: Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation
cites work: Inequalities with applications to percolation and reliability
cites work: Q3672870
cites work: The basic contact processes
cites work: Bernoulli percolation above threshold: An invasion percolation analysis
description: scientific article; zbMATH DE number 4003223