# Percolation of sites not removed by a random walker in $d$ dimensions

## TL;DR

This paper investigates how a random walk's site removal process affects percolation in high-dimensional lattices, revealing a sharp percolation threshold in dimensions above two and continuous behavior at the lower critical dimension.

## Contribution

It introduces a detailed study of correlated site percolation caused by random walks, identifying the percolation threshold and critical exponents across dimensions 2 to 6.

## Key findings

- Percolation probability approaches a step function as system size increases.
- Percolation threshold u_c is approximately 3 for dimensions 3 to 6.
- Correlation length diverges with exponents consistent with ν=2/(d-2).

## Abstract

How does removal of sites by a random walk lead to blockage of percolation? To study this problem of correlated site percolation, we consider a random walk (RW) of $N=uL^d$ steps on a $d$-dimensional hypercubic lattice of size $L^d$ (with periodic boundaries). We systematically explore dependence of the probability $\Pi_d(L,u)$ of percolation (existence of a spanning cluster) of sites not removed by the RW on $L$ and $u$. The concentration of unvisited sites decays exponentially with increasing $u$, while the visited sites are highly correlated -- their correlations decaying with the distance $r$ as $1/r^{d-2}$ (in $d>2$). Upon increasing $L$, the percolation probability $\Pi_d(L,u)$ approaches a step function, jumping from 1 to 0 when $u$ crosses a percolation threshold $u_c$ that is close to 3 for all $3\le d\le6$. Within numerical accuracy, the correlation length associated with percolation diverges with exponents consistent with $\nu=2/(d-2)$. There is no percolation threshold at the lower critical dimension of $d=2$, with the percolation probability approaching a smooth function $\Pi_2(\infty,u)>0$.