The lace expansion is a powerful and flexible method for understanding the critical scaling of several models of interest in probability, statistical mechanics, and combinatorics, above their upper critical dimensions. These models include the self-avoiding walk, lattice trees and lattice animals, percolation, oriented percolation, and the contact process. This volume provides a unified and extensive overview of the lace expansion and its applications to these models. Results include proofs of existence of critical exponents and construction of scaling limits. Often, the scaling limit is described in terms of super-Brownian motion.
1 Simple Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1<br />1.1 Asymptotic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1<br />1.2 Universality and Spread-Out Models . . . . . . . . . . . . . . . . . . . . . . . 5<br />2 The Self-Avoiding Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7<br />2.1 Asymptotic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7<br />2.2 Differential Inequalities and the Bubble Condition . . . . . . . . . . . 14<br />3 The Lace Expansion for the Self-Avoiding Walk . . . . . . . . . . . 19<br />3.1 Inclusion-Exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19<br />3.2 Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21<br />3.3 Laces and Resummation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24<br />3.4 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27<br />4 Diagrammatic Estimates for the Self-Avoiding Walk . . . . . . . 31<br />4.1 The Diagrammatic Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31<br />4.2 Proof of the Diagrammatic Estimates . . . . . . . . . . . . . . . . . . . . . . 33<br />5 Convergence for the Self-Avoiding Walk . . . . . . . . . . . . . . . . . . . 41<br />5.1 Random-Walk Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41<br />5.2 Convergence of the Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46<br />5.3 Finite Bubble vs Small Bubble . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54<br />5.4 Differential Equality and the Bubble Condition . . . . . . . . . . . . . . 54<br />6 Further Results for the Self-Avoiding Walk . . . . . . . . . . . . . . . . 57<br />6.1 The Self-Avoiding Walk in Dimensions d > 4 . . . . . . . . . . . . . . . . 57<br />6.2 A Self-Avoiding Walk in Dimension d = 1 . . . . . . . . . . . . . . . . . . 61<br />6.3 Nearest-Neighbour Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62<br />6.4 Networks of Self-Avoiding Walks . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7 Lattice Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67<br />7.1 Asymptotic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67<br />7.2 Differential Inequalities and the Square Condition . . . . . . . . . . . 71<br />8 The Lace Expansion for Lattice Trees . . . . . . . . . . . . . . . . . . . . . . 77<br />8.1 Expansion for Lattice Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77<br />8.2 Expansion for Lattice Animals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81<br />8.3 Diagrammatic Estimates for Lattice Trees . . . . . . . . . . . . . . . . . . 83<br />9 Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87<br />9.1 The Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87<br />9.2 Differential Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91<br />9.3 Differential Inequalities and the Triangle Condition . . . . . . . . . . 95<br />9.4 Proofs of the Differential Inequalities . . . . . . . . . . . . . . . . . . . . . . . 99<br />10 The Expansion for Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109<br />10.1 The Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109<br />10.2 The Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116<br />10.3 Diagrammatic Estimates for Percolation . . . . . . . . . . . . . . . . . . . . 121<br />11 Results for Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125<br />11.1 Critical Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125<br />11.2 The Critical Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129<br />11.3 The Incipient Infinite Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130<br />11.4 Percolation on Finite Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133<br />12 Oriented Percolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141<br />12.1 The Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141<br />12.2 The Infrared Bound and the Triangle Condition . . . . . . . . . . . . . 143<br />12.3 The Critical Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146<br />12.4 The Incipient Infinite Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147<br />13 Expansions for Oriented Percolation . . . . . . . . . . . . . . . . . . . . . . . 151<br />13.1 Inclusion-Exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152<br />13.2 Laces and Resummation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154<br />13.3 Application of the Percolation Expansion . . . . . . . . . . . . . . . . . . . 156<br />13.4 The Upper Critical Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157<br />13.5 Diagrams for Oriented Percolation . . . . . . . . . . . . . . . . . . . . . . . . . 158<br />14 The Contact Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161<br />14.1 The Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161<br />14.2 Approximation by Oriented Percolation . . . . . . . . . . . . . . . . . . . . 163<br />14.3 The Infrared Bound and the Triangle Condition . . . . . . . . . . . . . 164<br />14.4 Expansion for the Contact Process . . . . . . . . . . . . . . . . . . . . . . . . . 167<br />14.5 Diagrams for the Contact Process . . . . . . . . . . . . . . . . . . . . . . . . . 169 15 Branching Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171<br />15.1 A Mean-Field Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171<br />15.2 Branching Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172<br />15.3 Weakly Self-Avoiding Lattice Trees . . . . . . . . . . . . . . . . . . . . . . . . 180<br />16 Integrated Super-Brownian Excursion . . . . . . . . . . . . . . . . . . . . . 183<br />16.1 Moment Measures of Branching Random Walk . . . . . . . . . . . . . . 183<br />16.2 Critical Exponents and Generating Functions . . . . . . . . . . . . . . . 185<br />16.3 Construction of ISE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189<br />16.4 Lattice trees and ISE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194<br />16.5 Critical Percolation and ISE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197<br />17 Super-Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201<br />17.1 The Canonical Measure of SBM . . . . . . . . . . . . . . . . . . . . . . . . . . . 201<br />17.2 Moment Measures of the Canonical Measure . . . . . . . . . . . . . . . . 203<br />17.3 Critical Oriented Percolation and SBM . . . . . . . . . . . . . . . . . . . . . 207<br />17.4 The Critical Contact Process and SBM. . . . . . . . . . . . . . . . . . . . . 209<br />17.5 Lattice Trees and SBM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210<br />References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211<br />Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
