Dana Randall

Markov Chain Algorithms for Planar Lattice Structures

Consider the following Markov chain, whose states are all domino tilings of a 2n× 2n chessboard: starting from some arbitrary tiling, pick a 2×2 window uniformly at random. If the four squares appearing in this window are covered by two parallel dominoes, rotate the dominoes

Analyzing Glauber dynamics by comparison of Markov chains

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Dynamic TCP acknowledgement and other stories about e/(e-1)

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 two-dimensional 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.

Dynamic TCP Acknowledgment and Other Stories about e/(e - 1)

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 Saloff-Coste 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 time bounds for single-point update algorithms for problems such as generating planar tilings and random triangulations of convex polygons. ...

Sampling adsorbing staircase walks using a new Markov chain decomposition method

We present the first optimal randomized online algorithms for the TCP acknowledgment problem [5] and the Bahncard problem [7]. These problems are well-known to be generalizations of the classical online ski rental problem, however, they appeared to be harder. In this paper, we demonstrate that a number of online algorithms which have optimal competitive ratios of e/(e-1), including these, are fundamentally no more complex than ski rental. Our results also suggest a clear paradigm for solving ski rental-like problems.

Approximating the number of monomer-dimer coverings of a lattice

AbstractWe present the first optimal randomized online algorithms for the TCP acknowledgment problem [3] and the Bahncard problem [5]. These problems are well known to be generalizations of the classical online ski-rental problem, however, they appeared to be harder. In this paper we demonstrate that a number of online algorithms which have optimal competitive ratios of e/(e-1) , including these, are fundamentally no more complex than ski rental. Our results also suggest a clear paradigm for solving ski-rental-like problems.

Markov chain algorithms for planar lattice structures

Staircase walks are lattice paths from (0,0) to (2n,0) which take diagonal steps and which never fall below the x-axis. A path hitting the x-axis /spl kappa/ times is assigned a weight of /spl lambda//sup /spl kappa//, where /spl lambda/>0. A simple local Markov chain, which connects the state space and converges to the Gibbs measure (which normalizes these weights) is known to be rapidly mixing when /spl lambda/=1, and can easily be shown to be rapidly mixing when /spl lambda/<1. We give the first proof that this Markov chain is also mixing in the more interesting case of /spl lambda/>1, known in the statistical physics community as adsorbing staircase walks. The main new ingredient is a decomposition technique which allows us to analyze the Markov chain in pieces, applying different arguments to analyze each piece.

Slow mixing of glauber dynamics via topological obstructions

We study the problem of counting the number of coverings of ad-dimensional rectangular lattice by a specified number of monomers and dimers. This problem arises in several models in statistical physics, and has been widely studied. A classical technique due to Fisher, Kasteleyn, and Temperley solves the problem exactly in two dimensions when the number of monomers is zero (the dimer covering problem), but is not applicable in higher dimensions or in the presence of monomers. This paper presents the first provably polynomial-time approximation algorithms for computing the number of coverings with any specified number of monomers ind-dimensional rectangular lattices with periodic boundaries, for any fixed dimensiond, and in two-dimensional lattices with fixed boundaries. The algorithms are based on Monte Carlo simulation of a suitable Markov chain, and, in constrast to most Monte Carlo algorithms in statistical physics, have rigorously derived performance guarantees that do not rely on any assumptions. The method generalizes to counting coverings of any finite vertex-transitive graph, a class which includes most natural finite lattices with periodic boundary conditions.

Self-packing of centrally symmetric convex bodies in R 2

Consider the following Markov chain, whose states are all domino tilings of a 2n/spl times/2n chessboard: starting from some arbitrary tiling, pick a 2/spl times/2 window uniformly at random. If the four squares appearing in this window are covered by two parallel dominoes, rotate the dominoes in place. Repeat many times. This process is used in practice to generate a random tiling and is a key 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 two-dimensional lattices. The 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 van den Berg-Kesten-Reimer Inequality: A Review

Many local Markov chains based on Glauber dynamics are known to undergo a phase transition as a parameter λ of the system is varied. For independent sets on the 2-dimensional Cartesian lattice, the Gibbs distribution assigns each independent set a weight λ^[<i>I</i>], and the Markov chain adds or deletes a single vertex at each step, It is believed that there is a critical point λ<inf><i>c</i></inf> ≈ 3.79 such that for λ < λ<inf><i>c</i></inf>, local dynamics converge in polynomial time while for λ > λ<inf><i>c</i></inf> they require exponential time. We introduce a new method for showing slow mixing based on the presence or absence of certain topological obstructions in the independent sets. Using elementary arguments, we show that Glauber dynamics will be slow for sampling independent sets in 2 dimensions when λ ≥ 8.066, improving on the best known bound by a factor of 10. We also show they are slow on the torus when λ ≥ 6.183.