Discontinuity of the percolation density in one dimensional 1/|x-y|2 percolation models

M. Aizenman, C. M. Newman

Research output: Contribution to journalArticle

Abstract

We consider one dimensional percolation models for which the occupation probability of a bond -Kx,y, has a slow power decay as a function of the bond's length. For independent models - and with suitable reformulations also for more general classes of models, it is shown that: i) no percolation is possible if for short bonds Kx,y≦p<1 and if for long bonds Kx,y≦β/|x-y|2 with β≦1, regardless of how close p is to 1, ii) in models for which the above asymptotic bound holds with some β<∞, there is a discontinuity in the percolation density M (≡P) at the percolation threshold, iii) assuming also translation invariance, in the nonpercolative regime, the mean cluster size is finite and the two-point connectivity function decays there as fast as C(β, p)/|x-y|2. The first two statements are consequences of a criterion which states that if the percolation density M does not vanish then βM2>=1. This dichotomy resembles one for the magnetization in 1/|x-y|2 Ising models which was first proposed by Thouless and further supported by the renormalization group flow equations of Anderson, Yuval, and Hamann. The proofs of the above percolation phenomena involve (rigorous) renormalization type arguments of a different sort.

Original languageEnglish (US)
Pages (from-to)611-647
Number of pages37
JournalCommunications in Mathematical Physics
Volume107
Issue number4
DOIs
StatePublished - Dec 1986

Fingerprint

Discontinuity
discontinuity
dichotomies
flow equations
Dichotomy
Reformulation
Magnetization
Renormalization
Renormalization Group
occupation
Sort
Ising model
Ising Model
Model
Decay
magnetization
decay
Class

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Physics and Astronomy(all)
  • Mathematical Physics

Cite this

Discontinuity of the percolation density in one dimensional 1/|x-y|2 percolation models. / Aizenman, M.; Newman, C. M.

In: Communications in Mathematical Physics, Vol. 107, No. 4, 12.1986, p. 611-647.

Research output: Contribution to journalArticle

@article{2879e2da88554908b8e4ae33b6f8db11,
title = "Discontinuity of the percolation density in one dimensional 1/|x-y|2 percolation models",
abstract = "We consider one dimensional percolation models for which the occupation probability of a bond -Kx,y, has a slow power decay as a function of the bond's length. For independent models - and with suitable reformulations also for more general classes of models, it is shown that: i) no percolation is possible if for short bonds Kx,y≦p<1 and if for long bonds Kx,y≦β/|x-y|2 with β≦1, regardless of how close p is to 1, ii) in models for which the above asymptotic bound holds with some β<∞, there is a discontinuity in the percolation density M (≡P∞) at the percolation threshold, iii) assuming also translation invariance, in the nonpercolative regime, the mean cluster size is finite and the two-point connectivity function decays there as fast as C(β, p)/|x-y|2. The first two statements are consequences of a criterion which states that if the percolation density M does not vanish then βM2>=1. This dichotomy resembles one for the magnetization in 1/|x-y|2 Ising models which was first proposed by Thouless and further supported by the renormalization group flow equations of Anderson, Yuval, and Hamann. The proofs of the above percolation phenomena involve (rigorous) renormalization type arguments of a different sort.",
author = "M. Aizenman and Newman, {C. M.}",
year = "1986",
month = "12",
doi = "10.1007/BF01205489",
language = "English (US)",
volume = "107",
pages = "611--647",
journal = "Communications in Mathematical Physics",
issn = "0010-3616",
publisher = "Springer New York",
number = "4",

}

TY - JOUR

T1 - Discontinuity of the percolation density in one dimensional 1/|x-y|2 percolation models

AU - Aizenman, M.

AU - Newman, C. M.

PY - 1986/12

Y1 - 1986/12

N2 - We consider one dimensional percolation models for which the occupation probability of a bond -Kx,y, has a slow power decay as a function of the bond's length. For independent models - and with suitable reformulations also for more general classes of models, it is shown that: i) no percolation is possible if for short bonds Kx,y≦p<1 and if for long bonds Kx,y≦β/|x-y|2 with β≦1, regardless of how close p is to 1, ii) in models for which the above asymptotic bound holds with some β<∞, there is a discontinuity in the percolation density M (≡P∞) at the percolation threshold, iii) assuming also translation invariance, in the nonpercolative regime, the mean cluster size is finite and the two-point connectivity function decays there as fast as C(β, p)/|x-y|2. The first two statements are consequences of a criterion which states that if the percolation density M does not vanish then βM2>=1. This dichotomy resembles one for the magnetization in 1/|x-y|2 Ising models which was first proposed by Thouless and further supported by the renormalization group flow equations of Anderson, Yuval, and Hamann. The proofs of the above percolation phenomena involve (rigorous) renormalization type arguments of a different sort.

AB - We consider one dimensional percolation models for which the occupation probability of a bond -Kx,y, has a slow power decay as a function of the bond's length. For independent models - and with suitable reformulations also for more general classes of models, it is shown that: i) no percolation is possible if for short bonds Kx,y≦p<1 and if for long bonds Kx,y≦β/|x-y|2 with β≦1, regardless of how close p is to 1, ii) in models for which the above asymptotic bound holds with some β<∞, there is a discontinuity in the percolation density M (≡P∞) at the percolation threshold, iii) assuming also translation invariance, in the nonpercolative regime, the mean cluster size is finite and the two-point connectivity function decays there as fast as C(β, p)/|x-y|2. The first two statements are consequences of a criterion which states that if the percolation density M does not vanish then βM2>=1. This dichotomy resembles one for the magnetization in 1/|x-y|2 Ising models which was first proposed by Thouless and further supported by the renormalization group flow equations of Anderson, Yuval, and Hamann. The proofs of the above percolation phenomena involve (rigorous) renormalization type arguments of a different sort.

UR - http://www.scopus.com/inward/record.url?scp=0000847052&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000847052&partnerID=8YFLogxK

U2 - 10.1007/BF01205489

DO - 10.1007/BF01205489

M3 - Article

VL - 107

SP - 611

EP - 647

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 4

ER -