STRONG STATIONARY DUALITY FOR MÖBIUS MONOTONE MARKOV
CHAINS: EXAMPLES
Paweł Lorek
Ryszard Szekli
Abstract: We construct strong stationary dual chains for non-symmetric random walks on square
lattice, for random walks on hypercube and for some Ising models on the circle. The strong
stationary dual chains are all sharp and have the same state space as original chains. We use
Möbius monotonicity of these chains with respect to some natural orderings of the
corresponding state spaces. This method provides an alternative way to study mixing times
for studied models.
2000 AMS Mathematics Subject Classification: Primary: 60J10; Secondary: 06A06,
60G40.
Keywords and phrases: Markov chains, stochastic monotonicity, eigenvalues, Möbius
monotonicity, strong stationary duality, strong stationary times, separation distance, mixing
time, Ising model, hypercube.