Shape transition under excess self-intersections for transient random walk

TL;DR

This paper investigates a shape transition in transient random walks when the local times' excess norm crosses a critical value, and establishes a central limit theorem for the local times' norm in higher dimensions.

Contribution

It identifies a phase transition in the shape of the walk related to the excess q-norm of local times and proves a CLT for the q-norm in dimensions four and above.

Findings

Shape transition occurs at q_c(d)=d/(d-2)

Central limit theorem established for q-norm in dimensions ≥4

Provides new insights into local time behavior in transient walks

Abstract

We reveal a shape transition for a transient simple random walk forced to realize an excess $q$-norm of the local times, as the parameter $q$ crosses the value $q_c(d)=\frac{d}{d-2}$. Also, as an application of our approach, we establish a central limit theorem for the $q$-norm of the local times in dimension 4 or more.