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About this book

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.

Table of Contents

Frontmatter

2016 | OriginalPaper | Chapter

Chapter 1. Introduction

Alexander Barvinok

2016 | OriginalPaper | Chapter

Chapter 2. Preliminaries

Alexander Barvinok

2016 | OriginalPaper | Chapter

Chapter 3. Permanents

Alexander Barvinok

2016 | OriginalPaper | Chapter

Chapter 4. Hafnians and Multidimensional Permanents

Alexander Barvinok

2016 | OriginalPaper | Chapter

Chapter 5. The Matching Polynomial

Alexander Barvinok

2016 | OriginalPaper | Chapter

Chapter 6. The Independence Polynomial

Alexander Barvinok

2016 | OriginalPaper | Chapter

Chapter 7. The Graph Homomorphism Partition Function

Alexander Barvinok

2016 | OriginalPaper | Chapter

Chapter 8. Partition Functions of Integer Flows

Alexander Barvinok

Backmatter

Additional information

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