David B. Wilson

Exact sampling with coupled Markov chains and applications to statistical mechanics

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 algorithm itself. If the state space has a partial order that is preserved under the moves of the Markov chain, then the coupling is often particularly efficient. Using our approach, one can sample from the Gibbs distributions associated with various statistical mechanics models (including Ising, random-cluster, ice, and dimer) or choose uniformly at random from the elements of a finite distributive lattice.

Generating random spanning trees more quickly than the cover time

It is widely known how to generate random spanning trees of an undirected graph. Broder showed how at FOCS [6], and Aldous too found the algorithm [2]. Start at any vertex and do a simple random walk on the graph. Each time a vertex is first encountered, mark the edge from which it was discovered. When all the vertices are discovered, the marked edges form a random spanning tree. This algorithm is easy to code up, has small running time constants, and has a nice proof that it generates trees with the right probabilities. This paper gives a new algorithm for generating random spanning trees. It too is simple, easy to code up, and has nice proofs. The new algorithm also has the following advantages:

Fast exponentiation with precomputation

In several cryptographic systems, a fixed element g of a group (generally Z/qZ) is repeatedly raised to many different powers. In this paper we present a practical method of speeding up such systems. using precomputed values to reduce the number of multiplications needed. In practice this provides a substantial improvement over the level of performance that can be obtained using addition chains, and allows the computation of gn for n < N in O(log N/log log N) group multiplications. We also show how these methods can he parallelized, to compute powers in O(log log N) group multiplications with O(log N/ log log N) processors.

How to Get a Perfectly Random Sample from a Generic Markov Chain and Generate a Random Spanning Tree of a Directed Graph

A general problem in computational probability theory is that of generating a random sample from the state space of a Markov chain in accordance with the steady-state probability law of the chain. Another problem is that of generating a random spanning tree of a graph or spanning arborescence of a directed graph in accordance with the uniform distribution, or more generally in accordance with a distribution given by weights on the edges of the graph or digraph. This article gives algorithms for both of these problems, improving on earlier results and exploiting the duality between the two problems. Each of the new algorithms hinges on the recently introduced technique of coupling from the past or on the linked notions of loop-erased random walk and “cycle popping.”

Mixing times of lozenge tiling and card shuffling Markov chains

We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its equilibrium distribution. One application is to a class of Markov chains introduced by Luby, Randall, and Sinclair to generate random tilings of regions by lozenges. For an L X L region we bound the mixing time by O(L^4 log L), which improves on the previous bound of O(L^7), and we show the new bound to be essentially tight. In another application we resolve a few questions raised by Diaconis and Saloff-Coste, by lower bounding the mixing time of various card-shuffling Markov chains. Our lower bounds are within a constant factor of their upper bounds. When we use our methods to modify a path-coupling analysis of Bubley and Dyer, we obtain an O(n^3 log n) upper bound on the mixing time of the Karzanov-Khachiyan Markov chain for linear extensions.

Chip-Firing and Rotor-Routing on Directed Graphs

We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems.

EXCITED RANDOM WALK

A random walk on

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\mathbb{Z}^d

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is excited if the first time it visits a vertex there is a bias in one direction, but on subsequent visits to that vertex the walker picks a neighbor uniformly at random. We show that excited random walk on

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\mathbb{Z}^d