By I. N. Herstein

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1 The ﬁrst eigenvalue and phase transitions cribe the phase transitions, and this is now an active research ﬁeld. Further remarks are given at the end of Chapter 9. The next application we would like to mention is to Monte Carlo Markov chains. Monte Carlo Markov chains (MCMC) Consider a function with several local minima. The usual algorithms go at each step to a place that decreases the value of the function. 2). 2 The ﬁrst eigenvalue and random algorithm The MCMC algorithm avoids this by allowing the possibility of visiting other places, not only toward a local minimum.

It should also be pointed out that the sharp estimates introduced in Chapter 1 were obtained from the exponential rate in the W -metric with respect to some much more reﬁned metric ρ rather than the discrete one. Replacing Pk and P with Pk (t) and P (t), respectively, and then going to the operators, it is not diﬃcult to arrive at the following notion [cf. Chen (1994b; 1994a) for details]. 25. A coupling operator Ω is called ρ-optimal if Ω ρ(x1 , x2 ) = inf Ω ρ(x1 , x2 ) Ω for all x1 = x2 , where Ω varies over all coupling operators.

Kendall (1986) and M. Cranston (1991). In the case that x = y, the ﬁrst and the third couplings here are deﬁned to be the same as the second one. In probabilistic language, suppose that the original process is given by the stochastic diﬀerential equation √ dXt = 2 σ(Xt )dBt + b(Xt )dt, where (Bt ) is a Brownian motion. We want to construct a new process (Xt ), √ dXt = 2 σ (Xt )dBt + b (Xt )dt, on the same probability space, having the same distribution as that of (Xt ). Then, what we need is only to choose a suitable Brownian motion (Bt ).