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54
Exact Sampling with Coupled Markov Chains and Applications to Statistical Mechanics
, 1996
"... For many applications it is useful to sample from a finite set of objects in accordance with some particular distribution. One approach is to run an ergodic (i.e., irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has ..."
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Cited by 548 (13 self)
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For many applications it is useful to sample from a finite set of objects in accordance with some particular distribution. One approach is to run an ergodic (i.e., irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has run for M steps, with M sufficiently large, the distribution governing the state of the chain approximates the desired distribution. Unfortunately it can be difficult to determine how large M needs to be. We describe a simple variant of this method that determines on its own when to stop, and that outputs samples in exact accordance with the desired distribution. The method uses couplings, which have also played a role in other sampling schemes; however, rather than running the coupled chains from the present into the future, one runs from a distant point in the past up until the present, where the distance into the past that one needs to go is determined during the running of the al...
Analyzing Glauber Dynamics by Comparison of Markov Chains
 Journal of Mathematical Physics
, 1999
"... A popular technique for studying random properties of a combinatorial set is to design a Markov chain Monte Carlo algorithm. For many problems there are natural Markov chains connecting the set of allowable configurations which are based on local moves, or "Glauber dynamics." Typically the ..."
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Cited by 71 (16 self)
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A popular technique for studying random properties of a combinatorial set is to design a Markov chain Monte Carlo algorithm. For many problems there are natural Markov chains connecting the set of allowable configurations which are based on local moves, or "Glauber dynamics." Typically these single site update algorithms are difficult to analyze, so often the Markov chain is modified to update several sites simultaneously. Recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems. In this work we use the comparison technique of Diaconis and SaloffCoste to show that several of the natural single point update algorithms are efficient. The strategy is to relate the mixing rate of these algorithms to the corresponding nonlocal algorithms which have already been analyzed. This allows us to give polynomial bounds for single point update algorithms for problems such as generating planar tilings and random triangulations of c...
Completeness of the Bethe ansatz for the six and eightvertex models
 J. Stat. Phys
"... We discuss some of the difficulties that have been mentioned in the literature in connection with the Bethe ansatz for the sixvertex model and XXZ chain, and for the eightvertex model. In particular we discuss the “beyond the equator”, infinite momenta and exact complete string problems. We show h ..."
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Cited by 39 (1 self)
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We discuss some of the difficulties that have been mentioned in the literature in connection with the Bethe ansatz for the sixvertex model and XXZ chain, and for the eightvertex model. In particular we discuss the “beyond the equator”, infinite momenta and exact complete string problems. We show how they can be overcome and conclude that the coordinate Bethe ansatz does indeed give a complete set of states, as expected.
Theory of Computation of Multidimensional Entropy with an Application to the MonomerDimer Problem
 Advances of Applied Math
"... We outline the most recent theory for the computation of the exponential growth rate of the number of configurations on a multidimensional grid. As an application we compute the monomerdimer constant for the 2dimensional grid to 8 decimal digits, agreeing with the heuristic computations of Baxter ..."
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Cited by 30 (16 self)
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We outline the most recent theory for the computation of the exponential growth rate of the number of configurations on a multidimensional grid. As an application we compute the monomerdimer constant for the 2dimensional grid to 8 decimal digits, agreeing with the heuristic computations of Baxter, and for the 3dimensional grid with an error smaller than 1.35%.
Exact solution of the sixvertex model with domain wall boundary conditions. Critical line between . . .
, 2008
"... This is a continuation of the papers [4] of Bleher and Fokin and [6] of Bleher and Liechty, in which the large n asymptotics is obtained for the partition function Zn of the sixvertex model with domain wall boundary conditions in the disordered and ferroelectric phases, respectively. In the presen ..."
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Cited by 30 (7 self)
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This is a continuation of the papers [4] of Bleher and Fokin and [6] of Bleher and Liechty, in which the large n asymptotics is obtained for the partition function Zn of the sixvertex model with domain wall boundary conditions in the disordered and ferroelectric phases, respectively. In the present paper we obtain the large n asymptotics of Zn on the critical line between these two phases.
The Computational Complexity of Some Classical Problems from Statistical Physics
 In Disorder in Physical Systems
, 1990
"... this paper is to attempt to review and classify the di# culty of a range of problems, arising in the statistical mechanics of physical systems and to which I was introduced by J.M. Hammersley in the early sixties. Their common characteristics at the time were that they all seemed hard and there was ..."
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Cited by 25 (0 self)
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this paper is to attempt to review and classify the di# culty of a range of problems, arising in the statistical mechanics of physical systems and to which I was introduced by J.M. Hammersley in the early sixties. Their common characteristics at the time were that they all seemed hard and there was little existing mathematical machinery which was of much use in dealing with them. Twenty years later the situation has not changed dramatically; there do exist some mathematical techniques which appear to be tools in trade for this area, subadditive functions and transfer matrices for example, but they are still relatively few and despite a great deal of e#ort the number of exact answers which are known to the many problems posed is extremely small. Below we shall attempt to explain why this should be so by showing how the problems originally studied are special cases of a wide range of problems which can, in a well defined sense, be regarded as the most intractable enumeration problems that can sensibly be posed
How to Get an Exact Sample From a Generic Markov Chain and Sample a Random Spanning Tree From a Directed Graph, Both Within the Cover Time
 In Proceedings of the Seventh Annual ACMSIAM Symposium on Discrete Algorithms
, 1996
"... This paper shows how to obtain unbiased samples from an unknown Markov chain by observing it for O(T c ) steps, where T c is the cover time. This algorithm improves on several previous algorithms, and there is a matching lower bound. Using the techniques from the sampling algorithm, we also show how ..."
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Cited by 12 (2 self)
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This paper shows how to obtain unbiased samples from an unknown Markov chain by observing it for O(T c ) steps, where T c is the cover time. This algorithm improves on several previous algorithms, and there is a matching lower bound. Using the techniques from the sampling algorithm, we also show how to sample random directed spanning trees from a weighted directed graph, with arcs directed to a root, and probability proportional to the product of the edge weights. This tree sampling algorithm runs within 18 cover times of the associated random walk, and is more generally applicable than the algorithm of Broder and Aldous. 1 Introduction Random sampling of combinatorial objects has found numerous applications in computer science and statistics. Examples include approximate enumeration and estimation of the expected value of quantities. Usually there is a finite space X of objects, and a probability distribution ß on X , and we wish to sample object x 2 X with probability ß(x). One ef...
A little statistical mechanics for the graph theorist
, 2008
"... In this survey, we give a friendly introduction from a graph theory perspective to the qstate Potts model, an important statistical mechanics tool for analyzing complex systems in which nearest neighbor interactions determine the aggregate behavior of the system. We present the surprising equivalen ..."
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Cited by 11 (2 self)
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In this survey, we give a friendly introduction from a graph theory perspective to the qstate Potts model, an important statistical mechanics tool for analyzing complex systems in which nearest neighbor interactions determine the aggregate behavior of the system. We present the surprising equivalence of the Potts model partition function and one of the most renowned graph invariants, the Tutte polynomial, a relationship that has resulted in a remarkable synergy between the two fields of study. We highlight some of these interconnections, such as computational complexity results that have alternated between the two fields. The Potts model captures the effect of temperature on the system and plays an important role in the study of thermodynamic phase transitions. We discuss the equivalence of the chromatic polynomial and the zerotemperature antiferromagnetic partition function, and how this has led to the study of the complex zeros of these functions. We also briefly describe Monte Carlo simulations commonly used for Potts model analysis of complex systems. The Potts model has applications as widely varied as magnetism, tumor migration, foam behaviors, and social demographics, and we provide a sampling of these that also demonstrates some variations of the Potts model. We conclude with some current areas of investigation that emphasize graph theoretic approaches.