# Mathematical Statistical Physics, Volume 83

## 1st Edition

### Lecture Notes of the Les Houches Summer School 2005

**Series Volume Editors:**Anton Bovier François Dunlop Aernout Van Enter Frank Den Hollander Jean Dalibard

**eBook ISBN:**9780080479231

**Hardcover ISBN:**9780444528131

**Imprint:**Elsevier Science

**Published Date:**27th June 2006

**Page Count:**848

**View all volumes in this series:**Particle Physics beyond the Standard Model

## Table of Contents

Lecturers who Contributed to this volume

ÉCOLE D’ÉTÉ DE PHYSIQUE DES HOUCHES

École de Physique des Houches

Previous sessions

Organizers

Lecturers

Participants

Preface

Informal Seminars

Random Matrices and Determinantal Processes

1 Introduction

2 Point processes

3 Non-intersecting paths and the Aztec diamond

4 Asymptotics

5 The corner growth model

Acknowledgement

Some Recent Aspects of Random Conformally Invariant Systems

Overview

Conformal Random Geometry

1 Preamble

2 Introduction

3 Intersections of random walks

4 Mixing random & self-avoiding walks

5 Percolation clusters

Random Motions in Random Media

1 Introduction

2 RWRE

3 RCM

4 Back to RWRE

Effective criterion:

5 Diffusions in random environment

An Introduction to Mean Field Spin Glas Theory: Methods and Results

1 Introduction

2 The mean field ferromagnetic model. Convexity and cavity methods

3 The mean field spin glass model. Basic definitions

4 The interpolation method and its generalizations

5 The thermodynamic limit and the variational bounds

6 The Parisi representation for the free energy

7 Conclusion and outlook for future developments

Acknowledgments

Short-Range Spin Glasses: Selected Open Problems

1 Introduction

2 The Fortuin-Kasteleyn random cluster representation and phase transitions

3 Spin glass ground states and invasion percolation

4 Ground state multiplicity in the 2D EA spin glass

Acknowledgment

Computing the Number of Metastable States in Infinite-Range Models

Acknowledgments

Dynamics of Trap Models

1 Introduction

2 Definition of the Bouchaud trap model

2.2. Assumption 2.2.

2.3. Assumption 2.3.

2.1 Examples of trap models

2.2 Natural questions on trap models

2.3 References

3 The one-dimensional trap model

3.3 Aging results

3.4 Subaging results

3.5 Behaviour of the aging functions on different time scales

4 The trap model in dimension larger than one

4.3 Aging results

4.4 The coarse-graining procedure

4.5 References

5 The arcsine law as a universal aging scheme

6 Applications of the arcsine law

Appendix A Subordinators

Quantum Entropy and Quantum Information

1 Introduction

2 Rudiments of Classical Information Theory

3 Introduction to Quantum Information Theory

4 Open systems

5 Quantum entropy

6 Data compression in Quantum Information Theory

7 Quantum channels and additivity

Acknowledgments

Two Lectures on Iterative Coding and Statistical Mechanics

1 Introduction

2 Codes on graphs

3 A simple-minded bound and belief propagation

4 Density evolution a.k.a. distributional recursive equations

5 The area theorem and some general questions

6 Historical and bibliographical note

Evolution in Fluctuating Populations

1 Introduction

Acknowledgement

2 Some classical coalescent theory

4 Spatial structure and the Malécot formula

5 Spatial models

Multi-Scale Analysis of Population Models

1 Spatial diffusion models of population genetics

2 Duality and representation via coalescent processes

Elements of Nonequilibrium Statistical Mechanics

1 Elements of introduction

2 Elements of an H-theorem

3 Elements of heat conduction

4 Lagrangian approach

5 A little entropology

6 Closed systems

7 Open systems

8 Why is it useful?

9 What is missing — among other things?

Mathematical Aspects of the Abelian Sandpile Model

Open question

Gibbsianness and Non-Gibbsianness in Lattice Random Fields

1 Historical remarks and purpose of the course

2 Setup, notation, and basic notions

3 Probability kernels, conditional probabilities, and statistical mechanics

4 What it takes to be Gibbsian

5 What it takes to be non-Gibbsian

Acknowledgments

Simulation of Statistical Mechanics Models

1 Overview

2 The Swendsen–Wang algorithm: some recent progress

Acknowledgments

## Description

The proceedings of the 2005 les Houches summer school on Mathematical Statistical Physics give and broad and clear overview on this fast developing area of interest to both physicists and mathematicians.

## Key Features

- Introduction to a field of math with many interdisciplinary connections in physics, biology, and computer science
- Roadmap to the next decade of mathematical statistical mechanics
- Volume for reference years to come

## Readership

Libraries of mathematics and physics, Individual scientists

## Details

- No. of pages:
- 848

- Language:
- English

- Copyright:
- © Elsevier Science 2006

- Published:
- 27th June 2006

- Imprint:
- Elsevier Science

- eBook ISBN:
- 9780080479231

- Hardcover ISBN:
- 9780444528131

## About the Series Volume Editors

### Anton Bovier Series Volume Editor

### François Dunlop Series Volume Editor

### Aernout Van Enter Series Volume Editor

### Frank Den Hollander Series Volume Editor

### Jean Dalibard Series Volume Editor

Jean Dalibard works in the field of atomic physics and quantum optics. His recent activities is centered on the physics of cold quantum gases, in particular Bose-Einstein condensation.

### Affiliations and Expertise

Laboratoire Kastler Brossel, ENS, Paris, France