Scaling limit for the ant in high-dimensional labyrinths

TL;DR

This paper establishes conditions under which the scaling limit of random walks on large critical high-dimensional random graphs converges to Brownian motion on the Integrated Super-Brownian Excursion, advancing understanding of anomalous diffusion.

Contribution

It provides four natural sufficient conditions on critical graphs that guarantee the existence of a universal scaling limit for random walks in high dimensions.

Findings

Conditions verified for the trace of large critical branching random walks

Scaling limit proven for dimensions greater than 14

Supports conjecture of universality in high-dimensional critical structures

Abstract

We study here a detailed conjecture regarding one of the most important cases of anomalous diffusion, i.e the behavior of the "ant in the labyrinth". It is natural to conjecture (see [16] and [8]) that the scaling limit for random walks on large critical random graphs exists in high dimensions, and is universal. This scaling limit is simply the natural Brownian Motion on the Integrated Super-Brownian Excursion. We give here a set of four natural sufficient conditions on the critical graphs and prove that this set of assumptions ensures the validity of this conjecture. The remaining future task is to prove that these sufficient conditions hold for the various classical cases of critical random structures, like the usual Bernoulli bond percolation, oriented percolation, spread-out percolation in high enough dimension. In the companion paper [10], we do precisely that in a first case, the random walk on the trace of a large critical branching random walk. We verify the validity of these sufficient conditions and thus obtain the scaling limit mentioned above, in dimensions larger than 14.