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55
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 ..."
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Cited by 35 (8 self)
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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.
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 ..."
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Cited by 27 (6 self)
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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.
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 ..."
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Cited by 26 (11 self)
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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.
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 ..."
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Cited by 21 (14 self)
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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 ..."
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Cited by 20 (12 self)
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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 ..."
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Cited by 19 (3 self)
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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 ..."
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Cited by 19 (8 self)
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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 # #.
Random Colorings of a Cayley Tree
 IN CONTEMPORARY COMBINATORICS, B. BOLLOBAS, ED., BOLYAI SOCIETY MATHEMATICAL STUDIES, 2002
, 2000
"... Probability measures on the space of proper colorings of a Cayley tree (that is, an infinite regular connected graph with no cycles) are of interest not only in combinatorics but also in statistical physics, as states of the antiferromagnetic Potts model at zero temperature, on the "Bethe latti ..."
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Cited by 17 (1 self)
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Probability measures on the space of proper colorings of a Cayley tree (that is, an infinite regular connected graph with no cycles) are of interest not only in combinatorics but also in statistical physics, as states of the antiferromagnetic Potts model at zero temperature, on the "Bethe lattice". We concentrate on a particularly nice class of such measures which remain invariant under paritypreserving automorphisms of the tree. Making use of a correspondence with branching random walks on certain bipartite graphs, we determine when more than one such measure exists. The case of "uniform" measures is particularly interesting, and as it turns out, plays a special role. Some of the results herein are deducible from previous work of the authors and by members of the statistical physics community, but many are new. We hope that this work will serve as a helpful glimpse into the rapidly expanding intersection of combinatorics and statistical physics.