# Cutoff for random lifts of weighted graphs

## TL;DR

This paper establishes a cutoff phenomenon for the mixing time of random walks on random lifts of weighted graphs, even without reversibility, with the mixing time characterized by the entropy of the universal cover.

## Contribution

It proves a cutoff for the mixing time of random walks on non-reversible lifts of weighted graphs, linking it to the universal cover's entropy, and shows this is optimal among all lifts.

## Key findings

- Mixing time is w.h.p. proportional to log n divided by entropy h.
- Cutoff occurs at the mixing time h^{-1} log n.
- Results extend to non-reversible base graphs and regular graphs.

## Abstract

We prove a cutoff for the random walk on random $n$-lifts of finite weighted graphs, even when the random walk on the base graph $\mathcal{G}$ of the lift is not reversible. The mixing time is w.h.p. $t_{mix}=h^{-1}\log n$, where $h$ is a constant associated to $\mathcal{G}$, namely the entropy of its universal cover. Moreover, this mixing time is the smallest possible among all $n$-lifts of $\mathcal{G}$. In the particular case where the base graph is a vertex with $d/2$ loops, $d$ even, we obtain a cutoff for a $d$-regular random graph (as did Lubetzky and Sly in \cite{cutoffregular} with a slightly different distribution on $d$-regular graphs, but the mixing time is the same).