Random Walks and Physical Fields

✍ Scribed by Yves Le Jan

• Publisher: Springer Nature Switzerland
• Year: 2024
• Tongue: English
• Leaves: 188
• Series: Probability Theory and Stochastic Modelling 106
• Edition: 1
• Category: Library

✦ Synopsis

This book presents fundamental relations between random walks on graphs and field theories of mathematical physics. Such relations have been explored for several decades and remain a rapidly developing research area in probability theory.

The main objects of study include Markov loops, spanning forests, random holonomies, and covers, and the purpose of the book is to investigate their relations to Bose fields, Fermi fields, and gauge fields. The book starts with a review of some basic notions of Markovian potential theory in the simple context of a finite or countable graph, followed by several chapters dedicated to the study of loop ensembles and related statistical physical models. Then, spanning trees and Fermi fields are introduced and related to loop ensembles. Next, the focus turns to topological properties of loops and graphs, with the introduction of connections on a graph, loop holonomies, and Yang–Mills measure. Among the main results presented is an intertwining relation between merge-and-split generators on loop ensembles and Casimir operators on connections, and the key reflection positivity property for the fields under consideration.

Aimed at researchers and graduate students in probability and mathematical physics, this concise monograph is essentially self-contained. Familiarity with basic notions of probability, Poisson point processes, and discrete Markov chains are assumed of the reader.​

✦ Table of Contents

Preface
Acknowledgment
Contents
1 Markov Chains and Potential Theory on Graphs
1.1 Graphs and Markov Chains
1.2 Green Matrices and Hitting Distributions
1.3 Energy
1.4 Continuous Time Markov Chain and Semigroup
1.5 Feynman–Kac Formula
2 Loop Measures
2.1 Bridges and Based Loops Measures
2.2 Loops and Pointed Loops
2.3 Wreath Products *
2.4 Poissonian Loop Ensembles
2.5 Hitting Probabilities
3 Decompositions, Traces and Excursions
3.1 Traces of Markov Chains and Energy Decomposition
3.2 Excursion Theory
3.3 The One Point Case and the Excursion Measure
3.4 Application to Loop Hitting Distributions
4 Occupation Fields
4.1 Loop Occupation Fields
4.2 Occupation Fields and Bridge Measures
4.3 Occupation Fields of Loop Ensembles
4.4 Variation of the Energy Form
4.5 Example: Branching Processes with Immigration *
4.6 Moments and Polynomials of the Vertex Occupation Field *
5 Primitive Loops, Loop Clusters, and Loop Percolation
5.1 Primitive Discrete Loops
5.2 Loop Clusters Inequalities
5.3 Distribution of Loop Clusters in a Finite Graph
5.4 Computation Using Exit Distributions
5.5 Renewal Processes *
5.6 Bernoulli Percolation and Loop Percolation *
5.7 Loop Percolation on Zd with d2 *
6 The Gaussian Free Field
6.1 Definition and First Properties
6.2 Free Field and Occupation Fields
6.3 Wick Products *
7 Networks, Ising Model, Flows, and Configurations
7.1 Networks and Loop Ensembles
7.2 Eulerian Networks and Loop Ensembles for α=1
7.3 Merge and Split Markov Chains
7.4 Random Flows and Negative Split
7.5 Spin Networks and Loop Ensembles for α=12
7.6 FK-Ising Model
7.7 Kramers-Wannier Duality *
7.8 Networks and Configurations
8 Loop Erasure, Spanning Trees and Combinatorial Maps
8.1 Loop Erasure
8.2 Wilson's Algorithm and Random Spanning Forests
8.3 Reconstruction of L1
8.4 The Recurrent Case: Unrooted Spanning Trees
8.5 Dynamics on Spanning Trees
8.6 Configurations and Wilson Algorithm
8.7 Configurations and Combinatorial Maps *
9 Fock Spaces, Fermi Fields, and Applications
9.1 The Bosonic Fock Space
9.2 The Fermionic Fock Space
9.3 Another Approach to Grassmann Variables
9.4 The Transfer Current Theorem
9.5 Oriented Edges
9.6 Spanning Trees of Complete Graphs
9.7 Supersymmetry and Bridges Local Times
9.8 Interactions Between Trees and Loops Represented in Fock Space
10 Groups and Covers
10.1 Universal Cover and Fundamental Group
10.2 Geodesic Loops and Conjugacy Classes
10.3 Even Subsets
10.4 Homotopies and Geodesic Loops Distributions
10.5 Loop Homologies
10.6 Galois Coverings
11 Holonomies and Gauge Fields
11.1 Connections
11.2 Distribution of Non-Abelian Holonomies
11.3 Vector Fields Interacting with Gauge Fields: Gauge Invariance
11.4 Fields on the Covers: Decomposition and Interpretation
11.5 Random Connections, and Yang–Mills Measure
11.6 Split-and-Merge and Casimir Operators
11.7 Deformation and Marchenko-Migdal Equation
12 Reflection Positivity and Physical Space
12.1 Loops and Bosons
12.2 A Counter Example
12.3 Fermi Fields
12.4 Random Connections
12.5 Physical Hilbert Space and Time Shift
References
Index

✦ Subjects

Graphs, Markov Chains, Connections, Free field, Fock Spaces, Markov loop measures, Ising model, Spanning trees, Fermi fields, Casimir operator, Bose field