Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models

By T. Hara, Remco van der Hofstad, Gordon Slade. 2000

153 citations recorded by the source corpus.

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 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 $x \in {\mathbb{Z}^d}$, 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 $x \to \infty$. 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.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/0727e3e711c47559c01f9c535b130d1547451bd4

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In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Connectivity properties of random interlacement and intersection of random walks (2010, background): “ch that for any positive integer n and for any z 0,zn+1 ∈ Zd, X z1,...,zn∈Zd Yn i=0 min 1,|zi −zi+1| 2−d ( ≤ C|z 0−zn+1|2n+2−d if n < sd, = ∞ otherwise. (See, e.g. (1.38) of Proposition 1.7 in [4] for a proof of Lemma 1.) Lemma 1 gives bounds on n-fold convolutions of the probability that a ran”
The Alexander-Orbach conjecture holds in high dimensions (2008, methodology): “These facts alone suffice to control the behavior of the random walk and yield Theorem 1.1, as shown by Barlow, Járai, Kumagai and Slade [9].”
The Lace Expansion and its Applications (2006, background): “(b) [91] Fix any α > 0 (think of α small).”