Results 1  10
of
40
Phase Transitions on Nonamenable Graphs
, 2000
"... We survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph. Most attention is devoted to transitive graphs and trees. ..."
Abstract

Cited by 49 (8 self)
 Add to MetaCart
We survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph. Most attention is devoted to transitive graphs and trees.
Fast Mixing for Independent Sets, Colorings and Other Models on Trees
 IN PROCEEDINGS OF THE 15TH ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 2004
"... We study the mixing time of the Glauber dynamics for general spin systems on boundeddegree trees, including the Ising model, the hardcore model (independent sets) and the antiferromagnetic Potts model at zero temperature (colorings). We generalize a framework, developed in our recent paper [18] in ..."
Abstract

Cited by 36 (9 self)
 Add to MetaCart
We study the mixing time of the Glauber dynamics for general spin systems on boundeddegree trees, including the Ising model, the hardcore model (independent sets) and the antiferromagnetic Potts model at zero temperature (colorings). We generalize a framework, developed in our recent paper [18] in the context of the Ising model, for establishing mixing time O(n log n), which ties this property closely to phase transitions in the underlying model. We use this framework to obtain rapid mixing results for several models over a significantly wider range of parameter values than previously known, including situations in which the mixing time is strongly dependent on the boundary condition.
Glauber Dynamics on Trees and Hyperbolic Graphs
, 2001
"... We study discrete time Glauber dynamics for random configurations with local constraints (e.g. proper coloring, Ising and Potts models) on finite graphs with n vertices and of bounded degree. We show that the relaxation time (defined as the reciprocal of the spectral gap 1 \Gamma 2 ) for the dynami ..."
Abstract

Cited by 27 (11 self)
 Add to MetaCart
We study discrete time Glauber dynamics for random configurations with local constraints (e.g. proper coloring, Ising and Potts models) on finite graphs with n vertices and of bounded degree. We show that the relaxation time (defined as the reciprocal of the spectral gap 1 \Gamma 2 ) for the dynamics on trees and on certain hyperbolic graphs, is polynomial in n. For these hyperbolic graphs, this yields a general polynomial sampling algorithm for random configurations. We then show that if the relaxation time 2 satisfies 2 = O(n), then the correlation coefficient, and the mutual information, between any local function (which depends only on the configuration in a fixed window) and the boundary conditions, decays exponentially in the distance between the window and the boundary. For the Ising model on a regular tree, this condition is sharp. 1.
Optimal phylogenetic reconstruction
 In STOC ’06: Proceedings of the thirtyeighth annual ACM symposium on Theory of computing
, 2006
"... One of the major tasks of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. This problem is of critical importance in almost all areas of biology and has a very clear mathematical formulation. The evolutionary model is given by a Markov chain on the true evolution ..."
Abstract

Cited by 26 (6 self)
 Add to MetaCart
One of the major tasks of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. This problem is of critical importance in almost all areas of biology and has a very clear mathematical formulation. The evolutionary model is given by a Markov chain on the true evolutionary tree. Given samples from this Markov chain at the leaves of the tree, the goal is to reconstruct the evolutionary tree. It is crucial to minimize the number of samples, i.e., the length of genetic sequences, as it is constrained by the underlying biology, the price of sequencing etc. It is well known that in order to reconstruct a tree on n leaves, sequences of length Ω(log n) are needed. It was conjectured by M. Steel that for the CFN evolutionary model, if the mutation probability on all edges of the tree is less than p ∗ = ( √ 2 −1)/2 3/2 than the tree can be recovered from sequences of length O(log n). This was proven by the second author in the special case where the tree is “balanced”. The second author also proved that if all edges have mutation probability larger than p ∗ then the length needed is n Ω(1). This “phasetransition ” in the number of samples needed is closely related to the phase transition for the reconstruction problem (or extremality of free measure) studied extensively in statistical physics and probability.
A Phase Transition for a Random Cluster Model on Phylogenetic Trees
, 2004
"... We investigate a simple model that generates random partitions of the leaf set of a tree. Of particular interest is the reconstruction question: what number k of independent samples (partitions) are required to correctly reconstruct the underlying tree (with high probability)? We demonstrate a phase ..."
Abstract

Cited by 22 (14 self)
 Add to MetaCart
We investigate a simple model that generates random partitions of the leaf set of a tree. Of particular interest is the reconstruction question: what number k of independent samples (partitions) are required to correctly reconstruct the underlying tree (with high probability)? We demonstrate a phase transition for k as a function of the mutation rate, from logarithmic to polynomial dependence on the size of the tree. We also describe a simple polynomialtime tree reconstruction algorithm that applies in the logarithmic region. This model and the associated reconstruction questions are motivated by a Markov model for genomic evolution in molecular biology.
Inverting Random Functions II: Explicit Bounds for Discrete Maximum Likelihood Estimation, with Applications
 SIAM J. Discr. Math
, 2002
"... In this paper we study inverting randomfunctions under the maximumlikelihood estimation (MLE) criterion in the discrete setting. In particular, we consider how many independent evaluations of the random function at a particular element of the domain are needed for reliable reconstruction of that ele ..."
Abstract

Cited by 21 (13 self)
 Add to MetaCart
In this paper we study inverting randomfunctions under the maximumlikelihood estimation (MLE) criterion in the discrete setting. In particular, we consider how many independent evaluations of the random function at a particular element of the domain are needed for reliable reconstruction of that element. We provide explicit upper and lower bounds for MLE, both in the nonparametric and parametric setting, and give applications to cointossing and phylogenetic tree reconstruction.
Testing the Hypothesis of Common Ancestry
, 2002
"... this paper, we assess the arguments that have been made in the biological literature and discuss a methodology that has not been applied to this problem before ..."
Abstract

Cited by 20 (3 self)
 Add to MetaCart
this paper, we assess the arguments that have been made in the biological literature and discuss a methodology that has not been applied to this problem before
Robust Reconstruction on Trees is Determined By the Second Eigenvalue
, 2002
"... Consider information propagation from the root of infinite Bary tree, where each edge of the tree acts as an independent copy of some channel M . The reconstruction problem is solvable, if the n'th level of the tree contains a nonvanishing amount of information on the root of the tree, as n # ..."
Abstract

Cited by 18 (7 self)
 Add to MetaCart
Consider information propagation from the root of infinite Bary tree, where each edge of the tree acts as an independent copy of some channel M . The reconstruction problem is solvable, if the n'th level of the tree contains a nonvanishing amount of information on the root of the tree, as n # #.
Explicit isoperimetric constants and phase transitions in the randomcluster and Potts . . .
, 2000
"... The randomcluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results for the randomcluster model with cluster parameter q ≥ 1 are obtained. These include an explicit pointwise dynamical construction of randomclus ..."
Abstract

Cited by 18 (4 self)
 Add to MetaCart
The randomcluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results for the randomcluster model with cluster parameter q ≥ 1 are obtained. These include an explicit pointwise dynamical construction of randomcluster measures for arbitrary graphs, and for unimodular transitive graphs, lack of percolation for the free randomcluster measure at the lower critical value on nonamenable graphs, and a number of inequalities for the critical values. Some of these inequalities lead to considerations of isoperimetric constants in certain hyperbolic graphs, and the first nontrivial explicit calculations of such constants are obtained. Applications to the Potts model include Bernoullicity in the Z d case at all temperatures, and nonrobust phase transition in the case of nonamenable regular graphs.