The \(L^{2}\)-cutoff for reversible Markov processes
The concept of cutoff was introduced by D. Aldous and P. Diaconis in a series of papers to capture the fact that, for many ergodic Markov processes, the convergence to stationary behavior occurs over a period of time that is very short compared to the waiting time for the onset of such behavior. The aim of the present paper is twofold. The first one is to establish a criterion for the existence of an \(L^2\)--cutoff for families of Markov processes starting from specific initial distributions when the associated semigroup is normal. The second one is to derive formulas for the \(L^2\)--cutoff time sequence using spectral information. A typical result reads as follows: Let \(\Omega = \{ 0, 1, \ldots \}\) and \(K\) be the Markov kernel of the birth and death chain on \(\Omega \) with uniform birth rate \(p \in (0, 1/2)\), uniform death rate \(1 - p\) and \(K(0, 0) = 1 - p\). Let \((x_n)\) be a sequence of states in \(\Omega \). Then, the discrete time family of birth and death chains with respective starting states \(x_1, x_2, \ldots \) present an \(L^2\)--cutoff if and only if \(x_n\) tends to infinity. Moreover, if there is a cutoff then \[ t_n = \frac{\log (1 - p) - \log p}{- \log (4 p (1 - p))} x_n \] is a cutoff time sequence as \(n \rightarrow \infty \). The results are illustrated by several examples, including the Ehrenfest process and the biased \((p, q)\)--random walk on the non-negative integers.
scientific article