# Expansion of percolation critical points for Hamming graphs

## TL;DR

This paper derives an extended asymptotic expansion for the critical bond percolation point on Hamming graphs as the number of vertices grows, refining previous results and aiding understanding of phase transitions.

## Contribution

It provides a higher-order asymptotic expansion of the critical percolation point on Hamming graphs, extending prior work by one order and including the critical window width.

## Key findings

- Extended the asymptotic expansion of the critical point to one order higher.
- Identified the full asymptotic expansion for dimensions 4, 5, 6.
- Connected the results to critical percolation analysis on Hamming graphs.

## Abstract

The Hamming graph $H(d,n)$ is the Cartesian product of $d$ complete graphs on $n$ vertices. Let $m=d(n-1)$ be the degree and $V = n^d$ be the number of vertices of $H(d,n)$. Let $p_c^{(d)}$ be the critical point for bond percolation on $H(d,n)$. We show that, for $d \in \mathbb N$ fixed and $n \to \infty$, \begin{equation*} p_c^{(d)}= \dfrac{1}{m} + \dfrac{2d^2-1}{2(d-1)^2}\dfrac{1}{m^2} + O(m^{-3}) + O(m^{-1}V^{-1/3}), \end{equation*} which extends the asymptotics found in \cite{BorChaHofSlaSpe05b} by one order. The term $O(m^{-1}V^{-1/3})$ is the width of the critical window. For $d=4,5,6$ we have $m^{-3} = O(m^{-1}V^{-1/3})$, and so the above formula represents the full asymptotic expansion of $p_c^{(d)}$. In \cite{FedHofHolHul16a} \st{we show that} this formula is a crucial ingredient in the study of critical bond percolation on $H(d,n)$ for $d=2,3,4$. The proof uses a lace expansion for the upper bound and a novel comparison with a branching random walk for the lower bound. The proof of the lower bound also yields a refined asymptotics for the susceptibility of a subcritical Erd\H{o}s-R\'enyi random graph.