Upper bound on the expected size of the intrinsic ball

Artem Sapozhnikov

doi:10.1214/ecp.v15-1553

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.

Highlights

Y if there is an open path of at most n edges from x to y
We give a short proof of Theorem 1.2(i) from [5]
We show that the expected size of the intrinsic ball of radius r is at most C r if the susceptibility exponent γ is at most 1

Summary

Introduction

Y if there is an open path of at most n edges from x to y. AMS 2000 Subject classification: 60K35; 82B43 Keywords: Critical percolation; high-dimensional percolation; triangle condition; chemical distance; intrinsic ball. We give a short proof of Theorem 1.2(i) from [5]. We show that the expected size of the intrinsic ball of radius r is at most C r if the susceptibility exponent γ is at most 1.

Results

Conclusion