Glauber Dynamics On The Cycle Is Monotone

We study heat-bath Glauber dynamics for the ferromagnetic Ising model on a finite cycle (a graph where every vertex has degree two). We prove that the relaxation time $\tau_2$ is an increasing function of any of the couplings $J_{xy}$. We also prove some further inequalities, and obtain exact asymptotics for $\tau_2$ at low temperatures.