Cyclic Products and Optimal Traps in Cyclic Birth and Death Chains

A birth-death chain is a discrete-time Markov chain on the integers whose transition probabilities pi,j are non-zero if and only if |i - j| = 1. We consider birth-death chains whose birth probabilities pi,i+1 form a periodic sequence, so that pi,i+1 = pi mod m for some m and p0, ... , pm-1. The trajectory (X7,,)7,,=0,1,... of such a chain satisfies a strong law of large numbers and a central limit theo-rem. We study the effect of reordering the probabilities p0, ... , pm-1 on the velocity v = lim7,,& RARR;& INFIN; X7,,/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 (p0, . . . , pm-1), ex-actly (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 S 7. This conjecture is implied by a purely combinatorial conjecture that we think is of independent interest.