Results 1  10
of
66
The Markov Chain Monte Carlo method: an approach to approximate counting and integration
, 1996
"... In the area of statistical physics, Monte Carlo algorithms based on Markov chain simulation have been in use for many years. The validity of these algorithms depends crucially on the rate of convergence to equilibrium of the Markov chain being simulated. Unfortunately, the classical theory of stocha ..."
Abstract

Cited by 234 (13 self)
 Add to MetaCart
In the area of statistical physics, Monte Carlo algorithms based on Markov chain simulation have been in use for many years. The validity of these algorithms depends crucially on the rate of convergence to equilibrium of the Markov chain being simulated. Unfortunately, the classical theory of stochastic processes hardly touches on the sort of nonasymptotic analysis required in this application. As a consequence, it had previously not been possible to make useful, mathematically rigorous statements about the quality of the estimates obtained. Within the last ten years, analytical tools have been devised with the aim of correcting this deficiency. As well as permitting the analysis of Monte Carlo algorithms for classical problems in statistical physics, the introduction of these tools has spurred the development of new approximation algorithms for a wider class of problems in combinatorial enumeration and optimization. The “Markov chain Monte Carlo ” method has been applied to a variety of such problems, and often provides the only known efficient (i.e., polynomial time) solution technique.
Markov Chain Algorithms for Planar Lattice Structures
, 1995
"... Consider the following Markov chain, whose states are all domino tilings of a 2n x 2n chessboard: starting from some arbitrary tiling, pick a 2 x 2 window uniformly at random. If the four squares appearing in this window are covered by two parallel dominoes, rotate the dominoes 90° in place. Repeat ..."
Abstract

Cited by 93 (9 self)
 Add to MetaCart
Consider the following Markov chain, whose states are all domino tilings of a 2n x 2n chessboard: starting from some arbitrary tiling, pick a 2 x 2 window uniformly at random. If the four squares appearing in this window are covered by two parallel dominoes, rotate the dominoes 90° in place. Repeat many times. This process is used in practice to generate a random tiling, and is a widely used tool in the study of the combinatorics of tilings and the behavior of dimer systems in statistical physics. Analogous Markov chains are used to randomly generate other structures on various twodimensional lattices. This paper presents techniques which prove for the first time that, in many interesting cases, a small number of random moves suffice to obtain a uniform distribution.
The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
 In Survey in Combinatorics, 2005, volume 327 of London Mathematical Society Lecture Notes
, 2005
"... and matroids ..."
Homogeneous multivariate polynomials with the halfplane property
 Adv. in Appl. Math
"... A polynomial P in n complex variables is said to have the “halfplane property” (or Hurwitz property) if it is nonvanishing whenever all the variables lie in the open right halfplane. Such polynomials arise in combinatorics, reliability theory, electrical circuit theory and statistical mechanics. A ..."
Abstract

Cited by 39 (4 self)
 Add to MetaCart
A polynomial P in n complex variables is said to have the “halfplane property” (or Hurwitz property) if it is nonvanishing whenever all the variables lie in the open right halfplane. Such polynomials arise in combinatorics, reliability theory, electrical circuit theory and statistical mechanics. A particularly important case is when the polynomial is homogeneous and multiaffine: then it is the (weighted) generating polynomial of an runiform set system. We prove that the support (set of nonzero coefficients) of a homogeneous multiaffine polynomial with the halfplane property is necessarily the set of bases of a matroid. Conversely, we ask: For which matroids M does the basis generating polynomial P B(M) have the halfplane property? Not all matroids have the halfplane property, but we find large classes that do: all sixthrootofunity matroids, and a subclass of transversal (or cotransversal) matroids that we call “nice”. Furthermore, the class of matroids with the halfplane property is closed under minors, duality, direct sums, 2sums, series and parallel connection, fullrank matroid union, and some special cases of principal truncation, principal extension, principal cotruncation and principal coextension. Our positive results depend on two distinct (and apparently unrelated) methods for constructing polynomials with the halfplane property: a determinant construction (exploiting “energy” arguments), and a permanent construction (exploiting the Heilmann–Lieb theorem on matching polynomials). We conclude with a list of open questions. KEY WORDS: Graph, matroid, jump system, abstract simplicial complex, spanning tree, basis, generating polynomial, reliability polynomial, Brown–Colbourn conjecture,
Graph Colorings and Related Symmetric Functions: Ideas and Applications
, 1998
"... this paper we will report on further work related to this symmetric function ..."
Abstract

Cited by 35 (2 self)
 Add to MetaCart
this paper we will report on further work related to this symmetric function
Mathematical foundations of the Markov chain Monte Carlo method
 in Probabilistic Methods for Algorithmic Discrete Mathematics
, 1998
"... 7.2 was jointly undertaken with Vivek Gore, and is published here for the first time. I also thank an anonymous referee for carefully reading and providing helpful comments on a draft of this chapter. 1. Introduction The classical Monte Carlo method is an approach to estimating quantities that a ..."
Abstract

Cited by 30 (1 self)
 Add to MetaCart
7.2 was jointly undertaken with Vivek Gore, and is published here for the first time. I also thank an anonymous referee for carefully reading and providing helpful comments on a draft of this chapter. 1. Introduction The classical Monte Carlo method is an approach to estimating quantities that are hard to compute exactly. The quantity z of interest is expressed as the expectation z = ExpZ of a random variable (r.v.) Z for which some efficient sampling procedure is available. By taking the mean of some sufficiently large set of independent samples of Z, one may obtain an approximation to z. For example, suppose S = \Phi (x; y) 2 [0; 1] 2 : p i (x; y) 0; for all i \Psi<F12
On the CharneyDavis and NeggersStanley Conjectures
"... For a graded naturally labelled poset P, it is shown that the PEulerian polynomial ... counting linear extensions of P by their number of descents has symmetric and unimodal coefficient sequence, verifying the motivating consequence of the NeggersStanley conjecture on real zeroes for W (P, t) in t ..."
Abstract

Cited by 29 (3 self)
 Add to MetaCart
For a graded naturally labelled poset P, it is shown that the PEulerian polynomial ... counting linear extensions of P by their number of descents has symmetric and unimodal coefficient sequence, verifying the motivating consequence of the NeggersStanley conjecture on real zeroes for W (P, t) in these cases. The result is deduced from McMullen's gTheorem, by exhibiting a simplicial polytopal sphere whose hpolynomial is W (P, t). Whenever this...
Asymptotic enumeration of Latin rectangles
 J. Comb. Theory B
, 1990
"... A k X n Latin rectangle is a k X n matrix with entries from {1,2,..., n} such that no entry occurs more than once in any row or column. (Thus each row is a permutation of the integers 1,2,..., n.) Let L(k, n) be the number of k x n Latin rectangles. An outstanding problem is to determine the asympto ..."
Abstract

Cited by 24 (5 self)
 Add to MetaCart
A k X n Latin rectangle is a k X n matrix with entries from {1,2,..., n} such that no entry occurs more than once in any row or column. (Thus each row is a permutation of the integers 1,2,..., n.) Let L(k, n) be the number of k x n Latin rectangles. An outstanding problem is to determine the asymptotic value of L(k, n) as n — • oo, with k bounded by a suitable function of n. The first attack on this problem was made by Erdos and Kaplansky [1], who obtained the correct value for k — 0((log n) 3 / 2_e). The range of validity was later widened to A; = ofa 1 ^) by Yamamoto [8] and to k = ofo 1 / 2) by Stein [7]. We have obtained the correct value for k = o(n 6 / 7), and at the same time have sharpened the known approximations for fixed k> 4. Specifically, we have the following Theorem. THEOREM. Let k = 0(n x ~ 6) for some fixed 6> 0. Then L(M) J^nlr eMk{k mk n)) where
The repulsive lattice gas, the independentset polynomial, and the Lovász local lemma
, 2004
"... We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {px} if and only if the independent ..."
Abstract

Cited by 21 (7 self)
 Add to MetaCart
We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {px} if and only if the independentset polynomial for G is nonvanishing in the polydisc of radii {px}. Furthermore, we show that the usual proof of the Lovász local lemma — which provides a sufficient condition for this to occur — corresponds to a simple inductive argument for the nonvanishing of the independentset polynomial in a polydisc, which was discovered implicitly by Shearer [98] and explicitly by Dobrushin [37, 38]. We also present some refinements and extensions of both arguments, including a generalization of the Lovász local lemma that allows for “soft” dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternatingsign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.
Probabilistic bounds on the coefficients of polynomials with only real zeros
 J. Combin. Theory Ser. A
, 1997
"... The work of Harper and subsequent authors has shown that nite sequences (a 0;;an) arising from combinatorial problems are often such that the polynomial A(z): = P n k=0 akz k has only real zeros. Basic examples include rows from the arrays of binomial coe cients, Stirling numbers of the rst and sec ..."
Abstract

Cited by 20 (0 self)
 Add to MetaCart
The work of Harper and subsequent authors has shown that nite sequences (a 0;;an) arising from combinatorial problems are often such that the polynomial A(z): = P n k=0 akz k has only real zeros. Basic examples include rows from the arrays of binomial coe cients, Stirling numbers of the rst and second kinds, and Eulerian numbers. Assuming the ak are nonnegative, A(1)> 0 and that A(z) is not constant, it is known that A(z) has only real zeros i the normalized sequence (a 0=A(1);;an=A(1)) is the probability distribution of the Research supported in part by N.S.F. Grant MCS9404345 1 number of successes in n independent trials for some sequence of success probabilities. Such sequences (a 0;;an) are also known to be characterized by total positivity of the in nite matrix (ai,j) indexed by nonnegative integers i and j. This papers reviews inequalities and approximations for such sequences, called Polya frequency sequences which follow from their probabilistic representation. In combinatorial examples these inequalities yield a number of improvements of known estimates.