# Critical percolation on random regular graphs

## TL;DR

This paper analyzes the size, diameter, and mixing time of the largest component in percolated random regular graphs around the critical threshold, extending previous results and confirming theoretical predictions.

## Contribution

It extends known results on the largest component size in random regular graphs at the percolation threshold using a simple switching method.

## Key findings

- Largest component size is Θ(n^{2/3}) around the threshold
- Determines diameter and mixing time of the largest component
- Confirms predictions of Nachmias and Peres

## Abstract

We show that for all $d\in \{3,\ldots,n-1\}$ the size of the largest component of a random $d$-regular graph on $n$ vertices around the percolation threshold $p=1/(d-1)$ is $\Theta(n^{2/3})$, with high probability. This extends known results for fixed $d\geq 3$ and for $d=n-1$, confirming a prediction of Nachmias and Peres on a question of Benjamini. As a corollary, for the largest component of the percolated random $d$-regular graph, we also determine the diameter and the mixing time of the lazy random walk. In contrast to previous approaches, our proof is based on a simple application of the switching method.