The Scaling Limit of Lattice Trees Above Eight Dimensions

Abstract

<p>This work is concerned with the behavior of n-bond trees in a regular d dimensional lattice for large n. A lattice tree is, by definition, a connected cluster of bonds with no closed loops.</p> <p>The results presented herein are for a 'spread-out' model in d > 8; this model differs from the nearest-neighbour model in that the 'bonds' are chosen uniformly from {x,y ∈ Zᵈ : max₁≤ᵢ≤d |xᵢ-yᵢ| ≤ L} where L will be taken large enough for a variant of the lace expansion (as adapted in [HS3 & 4]) to converge.</p> <p>By way of comparison, it has been recently proved [HS1] via the lace expansion that, above 4 dimensions, n step self-avoiding walks on the hypercubic lattice Zᵈ converge (in distribution) to Brownian motion when space is scaled down by n½, and n tends to infinity.</p> <p>If we let tn(0,x) be the number of n bond trees connecting 0 and x, we can take its Fourier series tn(k):</p> <p>[equation removed]</p> <p>where k ∈ [-π,π]ᵈ. We prove that for L sufficiently large and d>8 for the spread-out model,</p> <p>[equation removed]</p> <p>where D is related to the mean radius of gyration. This would correspond in x space to scaling space down by n¼ where n is the size of a tree as measured by the number of bonds it contains.</p> <p>Similar calculations are carried out for trees connecting m points. The resulting distributions turn out to be exactly the characteristic functions of the measures D. Aldous conjectured in his 1993 J.S.P. paper regarding the embedding of random continuum trees in Rᵈ --which are themselves related to variant of super Brownian motion known as integrated super-Brownian excursion (ISE).</p>