## The $$L^{2}$$-cutoffs for reversible Markov chains.(English)Zbl 1374.60130

Summary: In this article, we considers reversible Markov chains of which $$L^{2}$$-distances can be expressed in terms of Laplace transforms. The cutoff of Laplace transforms was first discussed by the first author and L. Saloff-Coste [J. Funct. Anal. 258, No. 7, 2246–2315 (2010; Zbl 1192.60088)], while we provide here a completely different pathway to analyze the $$L^{2}$$-distance. Consequently, we obtain several considerably simplified criteria and this allows us to proceed advanced theoretical studies, including the comparison of cutoffs between discrete time lazy chains and continuous time chains. For an illustration, we consider product chains, a rather complicated model which could be involved to analyze using the method in [loc. cit.], and derive the equivalence of their $$L^{2}$$-cutoffs.

### MSC:

 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60J27 Continuous-time Markov processes on discrete state spaces

### Keywords:

product chains; cutoff phenomenon

Zbl 1192.60088
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