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Remco van der Hofstad

Publications and source records attributed to Remco van der Hofstad.

185 records · Page 11Linked to original sources
arXiv2004

Expansion in $n^{-1}$ for percolation critical values on the $n$-cube and $Z^n$: the first three terms

Let $p_c(\mathbb{Q}_n)$ and $p_c(\mathbb{Z}^n)$ denote the critical values for nearest-neighbour bond percolation on the $n$-cube $\mathbb{Q}_n = \{0,1\}^n$ and on $\Z^n$, respectively. Let $Ω= n$ for $\mathbb{G} = \mathbb{Q}_n$ and $Ω= 2n$ for $\mathbb{G} = \mathbb{Z}^n$ denote the degree of $\mathbb{G}$. We use the lace expansion to prove that for both $\mathbb{G} = \mathbb{Q}_n$ and $\mathbb{G} = \mathbb{Z}^n$, $p_c(\mathbb{G}) & = \cn^{-1} + \cn^{-2} + {7/2} \cn^{-3} + O(\cn^{-4}).$ This extends by two terms the result $p_c(\mathbb{Q}_n) = \cn^{-1} + O(\cn^{-2})$ of Borgs, Chayes, van der Hofstad, Slade and Spencer, and provides a simplified proof of a previous result of Hara and Slade for $\mathbb{Z}^n$.

math.PR↗
arXiv2000

A generalised inductive approach to the lace expansion

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 adaptation of the inductive method introduced by den Hollander and the authors. We give conditions under which the solution to the recursion relation behaves as a Gaussian, both in Fourier space and in terms of a local central limit theorem. These conditions are shown elsewhere to hold for sufficiently spread-out models of networks of self-avoiding walks in dimensions $d>4$, and for critical oriented percolation in dimensions $d+1>5$, providing a unified approach and an essential ingredient for a detailed analysis of the branching behaviour of these models.

math.PR↗
arXiv2000

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 the d-dimensional hyper cubic lattice having long finite-range connections, above their upper critical dimensions d=4 (self-avoiding walk), d=6 (percolation) and d=8 (trees and animals). The two-point functions for these models are respectively the generating function for self-avoiding walks from the origin to x, the probability of a connection from 0 to x, and the generating function for lattice trees or lattice animals containing 0 and x. We use the lace expansion to prove that for sufficiently spread-out models above the upper critical dimension, the two-point function of each model decays, at the critical point, as a multiple of $|x|^{2-d}$ as x goes to infinity. We use a new unified method to prove convergence of the lace expansion. The method is based on x-space methods rather than the Fourier transform. Our results also yield unified and simplified proofs of the bubble condition for self-avoiding walk, the triangle condition for percolation, and the square condition for lattice trees and lattice animals, for sufficiently spread-out models above the upper critical dimension.

math-ph↗
arXiv1999

Mean-field lattice trees

We introduce a mean-field model of lattice trees based on embeddings into $\Z^d$ of abstract trees having a critical Poisson offspring distribution. This model provides a combinatorial interpretation for the self-consistent mean-field model introduced previously by Derbez and Slade, and provides an alternate approach to work of Aldous. The scaling limit of the mean-field model is integrated super-Brownian excursion (ISE), in all dimensions. We also introduce a model of weakly self-avoiding lattice trees, in which an embedded tree receives a penalty $e^{-β}$ for each self-intersection. The weakly self-avoiding lattice trees provide a natural interpolation between the mean-field model ($β=0$), and the usual model of strictly self-avoiding lattice trees ($β=\infty$) which associates the uniform measure to the set of lattice trees of the same size.

math.PR↗