Upper bound on the expected size of intrinsic ball

TL;DR

This paper provides a concise proof that the expected size of an intrinsic ball in a high-dimensional percolation model is linearly bounded by its radius, under certain susceptibility conditions, confirming a key aspect of the Alexander-Orbach conjecture.

Contribution

It offers a simplified proof of an upper bound on the expected size of intrinsic balls, connecting susceptibility exponents and the triangle condition in high-dimensional percolation.

Findings

Expected size of intrinsic ball is at most proportional to its radius

The result holds if the susceptibility exponent is at most 1

The triangle condition implies the bound

Abstract

We give a short proof of Theorem 1.2 (i) from the paper "The Alexander-Orbach conjecture holds in high dimensions" by G. Kozma and A. Nachmias. We show that the expected size of the intrinsic ball of radius r is at most Cr if the susceptibility exponent is at most 1. In particular, this result follows if the so-called triangle condition holds.