On the mixing time of the Diaconis--Gangolli random walk on contingency tables over $\mathbb{Z}/ q \mathbb{Z}$

TL;DR

This paper analyzes the mixing time of the Diaconis--Gangolli random walk on contingency tables over finite cyclic groups, establishing cutoff phenomena and precise mixing time bounds under certain conditions.

Contribution

It provides the first rigorous analysis of the cutoff phenomenon for this random walk on contingency tables over rac{q}{ ext{Z}}, including explicit mixing time formulas.

Findings

Proves cutoff at rac{n^2}{4(1- rac{2 \u03c0}{q})} rac{ ext{log} n}{ ext{log} ext{log} n} conditions.

Establishes conditions on q for the cutoff to occur.

Provides explicit bounds on the mixing time for the random walk.

Abstract

The Diaconis--Gangolli random walk is an algorithm that generates an almost uniform random graph with prescribed degrees. In this paper, we study the mixing time of the Diaconis--Gangolli random walk restricted on $n\times n$ contingency tables over $\mathbb{Z}/q\mathbb{Z}$. We prove that the random walk exhibits cutoff at $\frac{n^2}{4(1- \cos{\frac{2 \pi}{q}})} \log n, $ when $\log q=o\left (\frac{\sqrt{\log n}}{\log \log n}\right )$.