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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…
THE LACE EXPANSION FOR SELF-AVOIDING WALK IN FIVE OR MORE DIMENSIONS
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
- École d'été de probabilités de Saint-Flour, G. Slade, J. Picard
- Mathematics
- 2006
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
- T. Hara, R. Hofstad, G. Slade
- Mathematics
- 28 November 2000
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
- R. van der Hofstad, G. Slade
- Mathematics
- 5 December 2000
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…
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