Aldous-type spectral gap results for the complete monomial group

TL;DR

This paper extends Aldous' spectral gap conjecture to the complete monomial group over a finite group, demonstrating that two related continuous-time random walks share the same spectral gap, thus revealing new symmetry properties.

Contribution

It proves that the spectral gaps of two natural random walks on the complete monomial group are equal, generalizing Aldous' spectral gap conjecture to this setting.

Findings

Spectral gaps of two random walks are equal

Extension of Aldous' conjecture to monomial groups

New symmetry property in group-based random walks

Abstract

Let us consider the continuous-time random walk on $G\wr S_n$, the complete monomial group of degree $n$ over a finite group $G$, as follows: An element in $G\wr S_n$ can be multiplied (left or right) by an element of the form \begin{itemize} \item $(u,v)_G:=(\mathbf{e},\dots,\mathbf{e};(u,v))$ with rate $x_{u,v}(\geq 0)$, or \item $(g)^{(w)}:=(\dots,\mathbf{e},\hspace*{-0.65cm}\underset{\substack{\uparrow\\w\text{th position}}}{g}\hspace*{-0.65cm},\mathbf{e},\dots;\mathbf{id})$ with rate $y_w\alpha_g\; (y_w> 0,\;\alpha_g=\alpha_{g^{-1}}\geq 0)$, \end{itemize} such that $\{(u,v)_G,\;(g)^{(w)}:x_{u,v}>0,\;y_w\alpha_g>0,\;1\leq u<v\leq n,\;g\in G,\;1\leq w\leq n\}$ generates $G\wr S_n$. We also consider the continuous-time random walk on $G\times\{1,\dots,n\}$ generated by one natural action of the elements $(u,v)_G,1\leq u<v\leq n$ and $(g)^{(w)},\;g\in G,1\leq w\leq n$ on $G\times\{1,\dots,n\}$ with the aforementioned rates. We show that the spectral gaps of the two random walks are the same. This is an analogue of the Aldous' spectral gap conjecture for the complete monomial group of degree $n$ over a finite group $G$.