| The Random-Cluster Model / Geoffrey Grimmett |
| Models of First-Passage Percolation / C. Douglas Howard |
| Relaxation Times of Markov Chains in Statistical Mechanics and Combinatorial Structures / Fabio Martinelli |
| Random Walks on Finite Groups / Laurent Saloff-Coste |
| Index |
| Introduction / 1： |
| A Motivating Example: the Assignment Problem / 1.1： |
| A Stalking Horse: the Partial Matching Problem / 1.2： |
| Organization of the Survey / 1.3： |
| Geometric Graphs and Local Weak Convergence / 2： |
| Geometric Graphs / 2.1： |
| <$>{\cal G}^\ast<$> as a Metric Space / 2.2： |
| Local Weak Convergence / 2.3： |
| The Standard Construction / 2.4： |
| A Prototype: The Limit of Uniform Random Trees / 2.5： |
| Maximal Weight Partial Matching on Random Trees / 3： |
| Weighted Matchings of Graphs in General / 3.1： |
| Our Case: Random Trees with Random Edge Weights / 3.2： |
| Two Obvious Guesses: One Right, One Wrong / 3.3： |
| Not Your Grandfather's Recursion / 3.4： |
| A Direct and Intuitive Plan / 3.5： |
| Characterization of the Limit of <$>B(T_n^{small})<$> / 3.6： |
| Characterization of the Limit of <$>B(T_n^{big})<$> / 3.7： |
| The Limit Theorem for Maximum Weight Partial Matchings / 3.8： |
| Closing the Loop: Another Probabilistic Solution of a Fixed-Point Equation / 3.9： |
| From Coupling to Stability - Thence to Convergence / 3.10： |
| Looking Back: Perspective on a Case Study / 3.11： |
| The Mean-Field Model of Distance / 4： |
| From Poisson Points in <$>{\op R}^{d}<$> to a Simple Distance Model / 4.1： |
| The Poisson Weighted Infinite Tree - or, the PWIT / 4.2： |
| The Cut-off Components of a Weighted Graph and a PWIT / 4.3： |
| The Minimum Spanning Forests of an Infinite Graph / 4.4： |
| The Average Length Per Vertex of the MSF of a PWIT / 4.5： |
| The Connection to Frieze's ζ(3) Theorem / 4.6： |
| Minimal Cost Perfect Matchings / 5： |
| A Natural Heuristic - Which Fails for a Good Reason / 5.1： |
| Involution Invariance and the Standard Construction / 5.2： |
| Involution Invariance and the Convergence of MSTs / 5.3： |
| A Heuristic That Works by Focusing on the Unknown / 5.4： |
| A Distributional Identity with a Logistic Solution / 5.5： |
| A Stochastic Process that Constructs a Matching / 5.6： |
| Calculation of a Limiting Constant: <$>\pi^{2}/6<$> / 5.7： |
| Passage from a PWIT Matching to a Kn Matching / 5.8： |
| Finally - Living Beyond One's Means / 5.9： |
| Problems in Euclidean Space / 6： |
| A Motivating Problem / 6.1： |
| Far Away Places and Their Influence / 6.2： |
| Euclidean Methods and Some Observations in Passing / 6.3： |
| Recurrence of Random Walks in Limits of Planar Graphs / 6.4： |
| Limitations, Challenges, and Perspectives / 7： |
| References |
| Potts and random-cluster processes |
| Random-cluster measures |
| Ising and Potts models |
| Random-cluster and Ising-Potts coupled |
| The limit as <$>q \downarrow 0<$> |
| Rank-generating functions |
| Infinite-volume random-cluster measures |
| Stochastic ordering |
| A differential formula |
| Conditional probabilities |
| Infinite-volume weak limits |
| Random-cluster measures on infinite graphs |
| The case q < 1 |
| Phase transition, the big picture |
| Infinite open clusters |
| First- and second-order phase transition |
| General results in d (≥ 2) dimensions |
| The subcritical phase, p < pc(q) |
| The supercritical phase, p > pc(q) |
| Near the critical point, p ≃ pc(q) |
| In two dimensions |
| Graphical duality |
| Value of the critical point |
| First-order phase transition |
| SLE limit when q ≤ 4 |
| On complete graphs and trees |
| On complete graphs / 7.1： |
| On trees and non-amenable graphs / 7.2： |
| Time-evolutions of random-cluster models / 8： |
| Reversible dynamics / 8.1： |
| Coupling from the past / 8.2： |
| Swendsen-Wang dynamics / 8.3： |
| The Basic Model and Some Fundamental Questions |
| Notation |
| The Time Constant |
| The Fundamental Processes of Hammersley and Welsh |
| About μ |
| Minimizing Paths |
| Asymptotic Shape and Shape Fluctuations |
| Shape Theorems for Standard FPP |
| About the Asymptotic Shape for Lattice FPP |
| FPP Based on Poisson Point Processes |
| Upper Bounds on Shape Fluctuations |
| Some Related Longitudinal Fluctuation Exponents |
| Monotonicity |
| Transversal Fluctuations and the Divergence of Shape Fluctuations |
| Transversal Fluctuation Exponents |
| Upper Bounds on <$>\xi<$> |
| Lower Bounds on <$>\chi<$> |
| Lower Bounds on <$>\xi<$> |
| Fluctuations for Other Related Models |
| Infinite Geodesics and Spanning Trees |
| Semi-Infinite Geodesics and Spanning Trees |
| Coalescence and Another Spanning Tree in 2 Dimensions |
| Doubly-Infinite Geodesics |
| Summary of Some Open Problems |
| Mixing times for reversible, continuous-time Markov chains |
| Analytic methods |
| Tensorization of the Poincaré and logarithmic Sobolev inequalities |
| Geometric tools |
| Comparison methods |
| Coupling methods and block dynamics |
| Statistical mechanics models in <$>{\op Z}^d<$> |
| Grand canonical Gibbs measures |
| Mixing conditions and absence of long-range order |
| Canonical Gibbs measures for lattice gases |
| The ferromagnetic Ising and Potts models |
| FK representation of Potts models |
| Antiferromagnetic models on an arbitrary graph: Potts and hard-core models |
| Model with random interactions |
| Unbounded spin systems |
| Ground states of certain quantum Heisenberg models as classical Gibbs measures |
| Glauber dynamics in <$>{\op Z}^d<$> |
| The dynamics in a finite volume |
| The dynamics in an infinite volume |
| Graphical construction |
| Uniform ergodicity and logarithmic Sobolev constant |
| Mixing property versus logarithmic Sobolev constant in <$>{\op Z}^d<$> |
| The auxiliary chain and sweeping out relations method |
| The renormalization group approach |
| The martingale method |
| The recursive analysis |
| Rapid mixing for unbounded spin systems |
| Torpid mixing in the phase coexistence region |
| Torpid mixing for the Ising model in <$>\Lambda \subset {\op Z}^{d}<$> with free boundary conditions |
| Interface driven mixing inside one phase |
| Torpid mixing for Potts model in <$>{\op Z}^d<$> |
| Glauber dynamics for certain random systems in <$>{\op Z}^d<$> |
| Combination of torpid and rapid mixing: the dilute Ising model |
| Relaxation to equilibrium for spin glasses |
| Glauber dynamics for more general structures |
| Glauber dynamics on trees and hyperbolic graphs |
| Glauber dynamics for the hard-core model |
| Cluster algorithms: the Swendsen-Wang dynamics for Potts models |
| Mixing time for conservative dynamics / 9： |
| Random transposition, Bernoulli-Laplace and symmetric simple exclusion / 9.1： |
| The asymmetric simple exclusion / 9.2： |
| The Kac model for the Boltzmann equation / 9.3： |
| Adsorbing staircase walks / 9.4： |
| Kawasaki dynamics for lattice gases / 10： |
| Diffusive scaling of the mixing time in the one-phase region / 10.1： |
| Background and Notation / 10.2： |
| Finite Markov Chains |
| Invariant Markov Chains on Finite Groups |
| Shuffling Cards and the Cut-off Phenomenon |
| Three Examples of Card Shuffling |
| Exact Computations |
| The Cut-off Phenomenon |
| Probabilistic Methods |
| Coupling |
| Strong Stationary Times |
| Spectrum and Singular Values |
| General Finite Markov Chains |
| The Random Walk Case |
| Lower Bounds |
| Eigenvalue Bounds Using Paths |
| Cayley Graphs |
| The Second Largest Eigenvalue |
| The Lowest Eigenvalue |
| Diameter Bounds, Isoperimetry and Expanders |
| Results Involving Volume Growth Conditions |
| Moderate Growth |
| Nilpotent Groups |
| Nilpotent Groups with many Generators / 7.3： |
| Representation Theory for Finite Groups |
| The General Set-up |
| Abelian Examples |
| Random Random Walks |
| Central Measures and Bi-invariant Walks |
| Characters and Bi-invariance |
| Random Transposition on the Symmetric Group |
| Walks Based on Conjugacy Classes of the Symmetric Group |
| Finite Classical Groups |
| Fourier Analysis for Non-central Measures / 9.5： |
| Comparison Techniques |
| The min-max Characterization of Eigenvalues |
| Comparing Dirichlet Forms Using Paths |
| Comparison for Non-symmetric Walks / 10.3： |
| The Random-Cluster Model / Geoffrey Grimmett |
| Models of First-Passage Percolation / C. Douglas Howard |
| Relaxation Times of Markov Chains in Statistical Mechanics and Combinatorial Structures / Fabio Martinelli |
図書
