David Aldous

Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists

Author(s): Aldous, DJ | Abstract: Consider N particles, which merge into clusters according to the following rule: a cluster of size x and a cluster of size y merge at (stochastic) rate K(x, y)/N, were AT is a specified rate kernel. This Marcus-Lushnikov model of stochastic coalescence and the underlying deterministic approximation given by the Smoluchowski coagulation equations have an extensive scientific literature. Some mathematical literature (Kingmans coalescent in population genetics; component sizes in random graphs) implicitly studies the special cases K(x, y) = 1 and K(x, y) = xy. We attempt a wide-ranging survey. General kernels are only now starting to be studied rigorously; so many interesting open problems appear.

Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem

We describe a simple one-person card game, patience sorting. Its analysis leads to a broad circle of ideas linking Young tableaux with the longest increasing subsequence of a random permutation via the Schensted correspondence. A recent highlight of this area is the work of Baik-Deift-Johansson which yields limiting probability laws via hard analysis of Toeplitz determinants.

SHUFFLING CARDS AND STOPPING-TIMES

EXAMPLE1. Top in at random shuffle. Consider the following method of mixing a deck of cards: the top card is removed and inserted into the deck at a random position. This procedure is repeated a number of times. The following argument should convince the reader that about n log n shuffles suffice to mix up n cards. The argument depends on following the bottom card of the deck. This card stays at the bottom until the first time (TI) a card is inserted below it. Standard calculations, reviewed below, imply this takes about n shuffles. As the shuffles continue, eventually a second card is inserted below the original bottom card (this takes about n/2 further shuffles). Consider the instant (T,) that a second card is inserted below the original bottom card. The two cards under the original bottom card are equally likely to be in relative order low-high or high-low. Similarly, the first time a h r d card is inserted below the original bottom card, each of the 6 possible orders of the 3 bottom cards is equally likely. Now consider the first time T,-, that the original bottom card comes up to the top. By an inductive argument, all (n l ) ! arrangements of the lower cards are equally likely. When the original bottom card is inserted at random, at time T = q,-, + 1, then all n! possible arrangements of the deck are equally likely.

Representations for partially exchangeable arrays of random variables

Consider an array of random variables (Xi,j), 1 ≤ i,j < ∞, such that permutations of rows or of columns do not alter the distribution of the array. We show that such an array may be represented as functions f(α, ξi, ηj, λi,j) of underlying i.i.d, random variables. This result may be useful in characterizing arrays with additional structure. For example, we characterize random matrices whose distribution is invariant under orthogonal rotation, confirming a conjecture of Dawid.

The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence

This survey describes a general approach to a class of problems that arise in combinatorial probability and combinatorial optimization. Formally, the method is part of weak convergence theory, but in concrete problems the method has a flavor of its own. A characteristic element of the method is that it often calls for one to introduce a new, infinite, probabilistic object whose local properties inform us about the limiting properties of a sequence of finite problems.

The random walk construction of uniform spanning trees and uniform labelled trees

A random walk on a finite graph can be used to construct a uniform random spanning tree. It is shown how random walk techniques can be applied to the study of several properties of the uniform random spanning tree: the proportion of leaves, the distribution of degrees, and the diameter.

Strong uniform times and finite random walks

There are several techniques for obtaining bounds on the rate of convergence to the stationary distribution for Markov chains with strong symmetry properties, in particular random walks on finite groups. An elementary method, strong uniform times, is often effective. We prove such times always exist, and relate this method to coupling and Fourier analysis.

PROBABILITY DISTRIBUTIONS ON CLADOGRAMS

By analogy with the theory surrounding the Ewens sampling formula in neutral population genetics, we ask whether there exists a natural one-parameter family of probability distributions on cladograms (“evolutionary trees”) which plays a central role in neutral evolutionary theory. Unfortunately the answer seems to be “no” - see Conjecture 2. But we can embed the two most popular models into an interesting family which we call “beta-splitting” models. We briefly describe some mathematical results about this family, which exhibits qualitatively different behavior for different ranges of the parameter β.

Hammersley's interacting particle process and longest increasing subsequences

SummaryIn a famous paper [8] Hammersley investigated the lengthLn of the longest increasing subsequence of a randomn-permutation. Implicit in that paper is a certain one-dimensional continuous-space interacting particle process. By studying a hydrodynamical limit for Hammersleys process we show by fairly “soft” arguments that limn′1/2ELn=2. This is a known result, but previous proofs [14, 11] relied on hard analysis of combinatorial asymptotics.

Asymptotics in the random assignment problem

SummaryWe show that, in the usual probabilistic model for the random assignment problem, the optimal cost tends to a limit constant in probability and in expectation. The method involves construction of an infinite limit structure, in terms of which the limit constant is defined. But we cannot improve on the known numerical bounds for the limit.