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The self-avoiding walk
The self-avoiding walk is a mathematical model with important applications in statistical mechanics and polymer science. This text provides a unified account of the rigorous results for the
Mean-field critical behaviour for percolation in high dimensions
AbstractThe triangle condition for percolation states that $$\sum\limits_{x,y} {\tau (0,x)\tau (0,y) \cdot \tau (y,0)} $$ is finite at the critical point, where τ(x, y) is the probability that the
This paper is a continuation of the companion paper [14], in which it was proved that the standard model of self-avoiding walk in five or more dimensions has the same critical behaviour as the simple
The Lace Expansion and its Applications
Simple Random Walk.- The Self-Avoiding Walk.- The Lace Expansion for the Self-Avoiding Walk.- Diagrammatic Estimates for the Self-Avoiding Walk.- Convergence for the Self-Avoiding Walk.- Further
Self-avoiding walk in five or more dimensions I. The critical behaviour
We use the lace expansion to study the standard self-avoiding walk in thed-dimensional hypercubic lattice, ford≧5. We prove that the numbercn ofn-step self-avoiding walks satisfiescn~Aμn, where μ is
Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models
We consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on ${\mathbb{Z}^d}$, having long finite-range connections, above their upper critical
Mean-Field Behaviour and the Lace Expansion
These lectures describe the lace expansion and its role in proving mean-field critical behaviour for self-avoiding walks, lattice trees and animals, and percolation, above their upper critical
On the upper critical dimension of lattice trees and lattice animals
We give a rigorous proof of mean-field critical behavior for the susceptibility (γ=1/2) and the correlation length (v=1/4) for models of lattice trees and lattice animals in two cases: (i) for the
The Scaling Limit of Lattice Trees in High Dimensions
Abstract:We prove that above eight dimensions the scaling limit of sufficiently spread-out lattice trees is the variant of super-Brownian motion known as integrated super-Brownian excursion (ISE), as
A generalised inductive approach to the lace expansion
Abstract. The lace expansion is a powerful tool for analysing the critical behaviour of self-avoiding walks and percolation. It gives rise to a recursion relation which we abstract and study using an