[ספר][B] Gibbs measures on Cayley trees

UA Rozikov - 2013‏ - books.google.com
The purpose of this book is to present systematically all known mathematical results on
Gibbs measures on Cayley trees (Bethe lattices). The Gibbs measure is a probability …

[ספר][B] Progress in high-dimensional percolation and random graphs

M Heydenreich, R Van der Hofstad - 2017‏ - Springer
This book focuses on percolation on high-dimensional lattices. We give a general
introduction to percolation, stating the main results and defining the central objects. We …

[ספר][B] Random walks on disordered media and their scaling limits

T Kumagai - 2014‏ - Springer
The main theme of these lecture notes is to analyze heat conduction on disordered media
such as fractals and percolation clusters by means of both probabilistic and analytic …

The Alexander-Orbach conjecture holds in high dimensions

G Kozma, A Nachmias - Inventiones mathematicae, 2009‏ - Springer
We examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-
field behavior has been established, namely when the dimension d is large enough or when …

Random subgraphs of finite graphs: I. The scaling window under the triangle condition

C Borgs, JT Chayes, R Van Der Hofstad… - Random Structures …, 2005‏ - Wiley Online Library
We study random subgraphs of an arbitrary finite connected transitive graph 𝔾 obtained by
independently deleting edges with probability 1− p. Let V be the number of vertices in 𝔾, and …

Phase transitions on nonamenable graphs

R Lyons - arxiv preprint math/9908177, 1999‏ - arxiv.org
arxiv:math/9908177v3 [math.PR] 16 Jun 2000 Page 1 arxiv:math/9908177v3 [math.PR] 16 Jun
2000 J. Math. Phys. 41 (2000), 1099–1126. Version of 30 March 2000 Phase Transitions on …

Arm exponents in high dimensional percolation

G Kozma, A Nachmias - Journal of the American Mathematical Society, 2011‏ - ams.org
We study the probability that the origin is connected to the sphere of radius $ r $(an arm
event) in critical percolation in high dimensions, namely when the dimension $ d $ is large …

The random-cluster model

G Grimmett - Probability on discrete structures, 2004‏ - Springer
The class of random-cluster models is a unification of a variety of stochastic processes of
significance for probability and statistical physics, including percolation, Ising, and Potts …

Gibbs measures on Cayley trees: results and open problems

UA Rozikov - Reviews in Mathematical Physics, 2013‏ - World Scientific
The purpose of this review paper is to present systematically all known results on Gibbs
measures on Cayley trees (Bethe lattices). There are about 150 papers which contain …

Random Subgraphs Of Finite Graphs: III. The Phase Transition For The n-Cube

C Borgs, JT Chayes, R Van der Hofstad, G Slade… - Combinatorica, 2006‏ - Springer
We study random subgraphs of the n-cube 0, 1 n, where nearest-neighbor edges are
occupied with probability p. Let pc (n) be the value of p for which the expected size of the …