Cyclic products and optimal traps in cyclic birth and death chains (Q6115507)

Summary: 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 \(m\leqslant 7\). This conjecture is implied by a purely combinatorial conjecture that we think is of independent interest.