SHARP THRESHOLDS FOR THE RANDOM-CLUSTER AND ISING MODELS

被引：14

作者：

Graham, Benjamin ^{[1
,2
]}

Grimmett, Geoffrey ^{[3
]}

机构：

[1] Univ British Columbia, Dept Math, Vancouver, BC V6T 1Z2, Canada

[2] Ecole Normale Super, DMA, F-75230 Paris 5, France

[3] Univ Cambridge, Ctr Math Sci, Stat Lab, Cambridge CB3 0WB, England

来源：

ANNALS OF APPLIED PROBABILITY | 2011年 / 21卷 / 01期

关键词：

Random-cluster model; Potts model; Ising model; percolation; box-crossing; influence; sharp threshold; colored random-cluster model; fuzzy Potts model; CRITICAL PROBABILITY; CRITICAL-BEHAVIOR; PHASE-TRANSITION; PERCOLATION; MAGNETIZATION; COEXISTENCE; UNIQUENESS;

D O I：

10.1214/10-AAP693

中图分类号：

O21 [概率论与数理统计]; C8 [统计学];

学科分类号：

020208 ; 070103 ; 0714 ;

摘要：

A sharp-threshold theorem is proved for box-crossing probabilities on the square lattice. The models in question are the random-cluster model near the self-dual point p(sd)(q) = root q/(1 + root q), the Ising model with external field, and the colored random-cluster model. The principal technique is an extension of the influence theorem for monotonic probability measures applied to increasing events with no assumption of symmetry.

引用

页码：240 / 265

页数：26

共 38 条

[1] DISCONTINUITY OF THE MAGNETIZATION IN ONE-DIMENSIONAL 1/[X-Y]2 ISING AND POTTS MODELS
AIZENMAN, M
CHAYES, JT
CHAYES, L
NEWMAN, CM
[J]. JOURNAL OF STATISTICAL PHYSICS, 1988, 50 (1-2) : 1 - 40
[2] ON THE CRITICAL-BEHAVIOR OF THE MAGNETIZATION IN HIGH-DIMENSIONAL ISING-MODELS
AIZENMAN, M
FERNANDEZ, R
[J]. JOURNAL OF STATISTICAL PHYSICS, 1986, 44 (3-4) : 393 - 454
[3] THE PHASE-TRANSITION IN A GENERAL-CLASS OF ISING-TYPE MODELS IS SHARP
AIZENMAN, M
BARSKY, DJ
FERNANDEZ, R
[J]. JOURNAL OF STATISTICAL PHYSICS, 1987, 47 (3-4) : 343 - 374
[4] On weak mixing in lattice models
Alexander, KS
[J]. PROBABILITY THEORY AND RELATED FIELDS, 1998, 110 (04) : 441 - 471
[5] [Anonymous], 1999, PERCOLATION
[6] Sharp phase transition and critical behaviour in 2D divide and colour models
Balint, Andras
Camia, Federico
Meester, Ronald
[J]. STOCHASTIC PROCESSES AND THEIR APPLICATIONS, 2009, 119 (03) : 937 - 965
[7] Baxter R., 2007, Exactly Solved Models in Statistical Mechanics
[8] STRICT INEQUALITY FOR CRITICAL-VALUES OF POTTS MODELS AND RANDOM-CLUSTER PROCESSES
BEZUIDENHOUT, CE
GRIMMETT, GR
KESTEN, H
[J]. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 1993, 158 (01) : 1 - 16
[9] The critical probability for random Voronoi percolation in the plane is 1/2
Bollobas, Bela
Riordan, Oliver
[J]. PROBABILITY THEORY AND RELATED FIELDS, 2006, 136 (03) : 417 - 468
[10] A short proof of the Harris-Kesten theorem
Bollobas, Bela
Riordan, Oliver
[J]. BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, 2006, 38 : 470 - 484

← 1 2 3 4 →