Simulated annealing and balance of recurrence order in time-inhomogeneous Markov chains

We consider Markov chains with transition probabilities proportional to powers of a small time-varying parameter. For such processes we introduce the notion of order of recurrence for the states and transitions and show that these orders satisfy a balance equation across every edge cutset in the graph of the Markov chain. As a special case, we consider the method of optimization by simulated annealing and show that there holds a "detailed balance" across every edge in the graph. This allows us to determine the necessary and sufficient condition on the cooling schedule in order to guarantee that the global minimum is hit with probability one.