THE RANDOM-CLUSTER MODEL
CONTENTS RANDOM-CLUSTER MEASURES INTRODUCTION RANDOM-CLUSTER MODEL ISING AND POTTS MODELS RANDOM-CLUSTER AND ISING/POTTS MODELS COUPLED THE LIMIT AS Q ?0 BASIC NOTATION MONOTONIC MEASURES STOCHASTIC ORDERING OF MEASURES POSITIVE ASSOCIATION IN?UENCE FOR MONOTONIC MEASURES SHARP THRESHOLDS FOR INCREASING EVENTS EXPONENTIAL STEEPNESS FUNDAMENTAL PROPERTIES CONDITIONAL PROBABILITIES DIFFERENTIAL FORMULAE AND SHARP THRESHOLDS COMPARISON INEQUALITIES PARTITION FUNCTIONS DOMINATION BY THE ISING MODEL SERIES AND PARALLEL LAWS NEGATIVE ASSOCIATION IN?NITE-VOLUME MEASURES IN?NITE GRAPHS BOUNDARY CONDITIONS IN?NITE-VOLUME WEAK LIMITS IN?NITE-VOLUME RANDOM-CLUSTER MEASURES UNIQUENESS VIA CONVEXITY OF PRESSURE POTTS AND RANDOM-CLUSTER MODELS ON IN?NITE GRAPHS PHASE TRANSITION THE CRITICAL POINT PERCOLATION PROBABILITIES UNIQUENESS OF RANDOM-CLUSTER MEASURES THE SUBCRITICAL PHASE EXPONENTIAL DECAY OF RADIUS EXPONENTIAL DECAY OF VOLUME THE SUPERCRITICAL PHASE AND THE WULFF CRYSTAL UNIQUENESS WHEN Q < 1 IN TWO DIMENSIONS PLANAR DUALITY THE VALUE OF THE CRITICAL POINT FIRST-ORDER PHASE TRANSITION GENERAL LATTICES IN TWO DIMENSIONS SQUARE, TRIANGULAR, AND HEXAGONAL LATTICES STOCHASTIC L¨OWNER EVOLUTIONS DUALITY IN HIGHER DIMENSIONS SURFACES AND PLAQUETTES BASIC PROPERTIES OF SURFACES A CONTOUR REPRESENTATION POLYMER MODELS DISCONTINUOUS PHASE TRANSITION FOR LARGE Q DOBRUSHIN INTERFACES PROBABILISTIC AND GEOMETRIC PRELIMINARIES THE LAW OF THE INTERFACE GEOMETRY OF INTERFACES EXPONENTIAL BOUNDS FOR GROUP PROBABILITIES LOCALIZATION OF INTERFACE DYNAMICS OF RANDOM-CLUSTER MODELS TIME-EVOLUTION OF THE RANDOM-CLUSTER MODEL GLAUBER DYNAMICS GIBBS SAMPLER COUPLING FROM THE PAST SWENDSEN–WANG DYNAMICS COUPLED DYNAMICS ON A ?NITE GRAPH BOX DYNAMICS WITH BOUNDARY CONDITIONS COUPLED DYNAMICS ON THE IN?NITE LATTICE SIMULTANEOUS UNIQUENESS FLOWS IN POISSON GRAPHS POTTS MODELS AND ?OWS FLOWS IN THE ISING MODEL EXPONENTIAL DECAY FOR THE ISING MODEL THE ISING MODEL IN FOUR AND MORE DIMENSIONS 10 ON OTHER GRAPHS MEAN-?ELD THEORY ON COMPLETE GRAPHS MAIN RESULTS FOR THE COMPLETE GRAPH THE FUNDAMENTAL PROPOSITION THE SIZE OF THE LARGEST COMPONENT PROOFS OF MAIN RESULTS FOR COMPLETE GRAPHS THE NATURE OF THE SINGULARITY LARGE DEVIATIONS ON A TREE 1010 THE CRITICAL POINT FOR A TREE 1011 (NON-)UNIQUENESS OF MEASURES ON TREES 1012 ON NON-AMENABLE GRAPHS 11 GRAPHICAL METHODS FOR SPIN SYSTEMS RANDOM-CLUSTER REPRESENTATIONS THE POTTS MODEL THE ASHKIN–TELLER MODEL THE DISORDERED POTTS FERROMAGNET THE EDWARDS–ANDERSON SPIN-GLASS MODEL THE WIDOM–ROWLINSON LATTICE GAS