The Cutoff Phenomenon for Random Birth and Death Chains

TL;DR

This paper investigates the cutoff phenomenon in random birth and death Markov chains with a given stationary distribution, analyzing a block sampling method to understand typical chain behavior and comparing cutoff properties.

Contribution

It introduces a block sampler algorithm for generating random birth and death chains and compares cutoff phenomena between typical and special chains with the same stationary distribution.

Findings

Cutoff behavior is similar between random and special birth and death chains.

The block sampler effectively generates representative chains for analysis.

Results suggest typical chains exhibit cutoff under certain conditions.

Abstract

For any distribution $\pi$ with support equal to $[n] = \{1, 2,..., n \}$, we study the set $\mathcal{A}_{\pi}$ of tridiagonal stochastic matrices $K$ satisfying $\pi(i) K[i,j] = \pi(j) K[j,i]$ for all $i, j \in [n]$. These matrices correspond to birth and death chains with stationary distribution $\pi$. We study matrices $K$ drawn uniformly from $\mathcal{A}_{\pi}$, following the work of Diaconis and Wood on the case $\pi(i) = \frac{1}{n}$. We analyze a `block sampler' version of their algorithm for drawing from $\mathcal{A}_{\pi}$ at random, and use results from this analysis to draw conclusions about typical matrices. The main result is a soft argument comparing cutoff for sequences of random birth and death chains to cutoff for a special family of birth and death chains with the same stationary distributions.