Backbone scaling limit of the high-dimensional IIC: extended version

TL;DR

This paper determines the scaling limit of the backbone in high-dimensional incipient infinite clusters, showing it converges to Brownian motion in finite-range cases and to stable motion in long-range cases, using a new lace expansion technique.

Contribution

It introduces a novel lace expansion method that tracks pivotal bonds to identify the backbone's scaling limit in high-dimensional IICs.

Findings

Backbone scaling limit is Brownian motion in finite-range setting.

Backbone scaling limit is stable motion in long-range setting.

Lace expansion technique effectively tracks pivotal bonds.

Abstract

We identify the scaling limit of the backbone of the high-dimensional incipient infinite cluster (IIC), both in the finite-range and the long-range setting. In the finite-range setting, this scaling limit is Brownian motion, in the long-range setting, it is a stable motion. The proof relies on a novel lace expansion that keeps track of the number of pivotal bonds.