Combinatorics and Complexity of Partition Functions

Partition functions arise in combinatorics and related problems of statistical physics as they encode in a succinct way the combinatorial structure of complicated systems. The main focus of the book is on efficient ways to compute (approximate) various partition functions, such as permanents, hafnians and their higher-dimensional versions, graph and hypergraph matching polynomials, the independence polynomial of a graph and partition functions enumerating 0-1 and integer points in polyhedra, which allows one to make algorithmic advances in otherwise intractable problems.

The book unifies various, often quite recent, results scattered in the literature, concentrating on the three main approaches: scaling, interpolation and correlation decay. The prerequisites include moderate amounts of real and complex analysis and linear algebra, making the book accessible to advanced math and physics undergraduates.

Contains an exposition of recent results Demonstrates a unified approach to hard algorithmic problems Provides an easy to read introduction to statistical physics phenomena Includes supplementary material: sn.pub/extras

Auteur

Alexander Barvinok is a professor of mathematics at the University of Michigan in Ann Arbor, interested in computational complexity and algorithms in algebra, geometry and combinatorics. The reader might be familiar with his books A Course in Convexity (AMS, 2002) and Integer Points in Polyhedra (EMS, 2008)

Contenu

Chapter I. Introduction.- Chapter II. Preliminaries.- Chapter III. Permanents.- Chapter IV. Hafnians and Multidimensional Permanents.- Chapter V. The Matching Polynomial.- Chapter VI. The Independence Polynomial.- Chapter VII. The Graph Homomorphism Partition Function.- Chapter VIII. Partition Functions of Integer Flows.- References.- Index.