Logarithmic components of the vacant set for random walk on a discrete torus

TL;DR

This paper studies the structure of the unvisited points by a random walk on a high-dimensional discrete torus, revealing that some components are logarithmic in size, emphasizing the importance of the giant component's size threshold.

Contribution

It demonstrates that the complement of the giant component in the vacant set contains segments of logarithmic size, highlighting the significance of the constant c_0 in defining the giant component.

Findings

Some components of the vacant set are of logarithmic size.

The giant component's size threshold c_0 is crucial.

The complement of the giant component contains small segments.

Abstract

This work continues the investigation, initiated in a recent work by Benjamini and Sznitman, of percolative properties of the set of points not visited by a random walk on the discrete torus (Z/NZ)^d up to time uN^d in high dimension d. If u>0 is chosen sufficiently small it has been shown that with overwhelming probability this vacant set contains a unique giant component containing segments of length c_0 log N for some constant c_0 > 0, and this component occupies a non-degenerate fraction of the total volume as N tends to infinity. Within the same setup, we investigate here the complement of the giant component in the vacant set and show that some components consist of segments of logarithmic size. In particular, this shows that the choice of a sufficiently large constant c_0 > 0 is crucial in the definition of the giant component.