Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

This book is an introduction to the modern approach to the theory of Markov chains. The main goal of this approach is to determine the rate of convergence of a Markov chain to the stationary distribution as a function of the size and geometry of the state space. The authors develop the key tools for estimating convergence times, including coupling, strong stationary times, and spectral methods. Whenever possible, probabilistic methods are emphasized. The book includes many examples and provides brief introductions to some central models of statistical mechanics. Also provided are accounts of random walks on networks, including hitting and cover times, and analyses of several methods of shuffling cards. As a prerequisite, the authors assume a modest understanding of probability theory and linear algebra at an undergraduate level. ""Markov Chains and Mixing Times"" is meant to bring the excitement of this active area of research to a wide audience.

Markov Chains and Mixing Times

Sphere Packings, Lattices and Groups

Generating a random permutation with random transpositions

Consider the problem of approximately counting weighted independent sets of a graph G with activity λ, i.e., where the weight of an independent set I is λ|I|. We present a novel analysis yielding a deterministic approximation scheme which runs in polynomial time for any graph of maximum degree Δ and λ

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Counting independent sets up to the tree threshold

Introduction Extremal properties of the Wulff functional Limit theorems Surface tension Large contours Proof of the main results.

Wulff Construction: A Global Shape from Local Interaction

An interactionU is called a completely analytical (CA) interaction, if it satisfies one of 12 given conditions formulated in terms of analyticity properties of the partition functions Zv(u), or correlation decay, or truncated correlation bounds, or asymptotic behavior of ln Zv(u), v→∞. The 12 conditions are presented, together with part of the proof of their equivalence. The main result of the paper is that each condition is constructive in the following sense: instead of checking it in all finite volumesv⊂ℤv, it is enough to consider only (a finite amount of) volumes with restricted size. In particular, the partition functions Z
 v
 (u+ũ) for the complex perturbationsu+ũ ofu do not vanish for all Vℤ
 v
 and all Ũ with ∥Ũ∥<ɛ, provided this is true only forv with diam v⩽C(ɛ) and ∥Ũ∥<ɛ′ (but withɛ<ɛ′).

Completely Analytical Interactions: Constructive Description

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On a Topological Generalization of a Theorem of Tverberg

We show the existence of a first-order phase transition in thev-dimensional Potts model forv≧2, when the number of states of a single spin is big enough. Low-temperature pure phases are proved to survive up to the critical temperature. Also the existence of a first-order transition in thev-dimensional Potts gauge model,v≧3, is obtained if the underlying gauge group is finite but large.

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First-order phase transitions in large entropy lattice models

We develop a new analysis of the order-disorder transition in ferromagnetic Potts models for large numberq of spin states We use the Pirogov-Sinai theory which we adapt to the Fortuin-Kasteleyn representation of the models This theory applies in a rather direct way in our approach and leads to a system of non-interacting contours with small activities As a consequence, simpler and more natural techniques are found, allowing us to recover previous results on the bulk properties of the model (which then extend to non-integer values ofq) and to deal with non-translation invariant boundary conditions This will be applied in a second part of this work to study the behaviour of the interfaces at the transition point

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Senya Shlosman

Papers

Wulff Construction: A Global Shape from Local Interaction

Completely Analytical Interactions: Constructive Description

On a Topological Generalization of a Theorem of Tverberg

First-order phase transitions in large entropy lattice models

Interfaces in the Potts model I: Pirogov-Sinai theory of the Fortuin-Kasteleyn representation