Cyclic products and optimal traps in cyclic birth and death chains

TL;DR

This paper investigates how reordering periodic transition probabilities in cyclic birth-death chains affects the chain's asymptotic velocity, revealing that most configurations can produce multiple distinct speeds and proposing an optimal ordering conjecture.

Contribution

It characterizes the impact of probability reordering on the chain's velocity, proves the conjecture for small periods, and introduces a combinatorial conjecture of independent interest.

Findings

Reordering probabilities can produce multiple distinct asymptotic speeds.

For almost all probability configurations, exactly half of the permutations yield unique speeds.

The minimal speed ordering is conjectured and proven for periods up to 7.

Abstract

A birth-death chain is a discrete-time Markov chain on the integers whose transition probabilities $p_{i,j}$ are non-zero if and only if $|i-j|=1$. We consider birth-death chains whose birth probabilities $p_{i,i+1}$ form a periodic sequence, so that $p_{i,i+1}=p_{i \mod m}$ for some $m$ and $p_0,\ldots,p_{m-1}$. The trajectory $(X_n)_{n=0,1,\ldots}$ of such a chain satisfies a strong law of large numbers and a central limit theorem. We study the effect of reordering the probabilities $p_0,\ldots,p_{m-1}$ on the velocity $v=\lim_{n\to\infty} X_n/n$. The sign of $v$ is not affected by reordering, but its magnitude in general is. We show that for Lebesgue almost every choice of $(p_0,\ldots,p_{m-1})$, exactly $(m-1)!/2$ distinct speeds can be obtained by reordering. We make an explicit conjecture of the ordering that minimises the speed, and prove it for all $m\leq 7$. This conjecture is implied by a purely combinatorial conjecture that we think is of independent interest.