The max-`ℓ^2` mixing of reversible Markov chains
Guan-Yu Chen
Department of Applied Mathematics
National Chiao Tung University
gychen@math.nctu.edu.tw
The `ℓ^2`-distance is one frequently used measurement to analyze the convergence of Markov chains to their stationarity. For reversible Markov chains, their `ℓ^2`-distances can be formulated by the spectral information of their transition matrices. The corresponding mixing time and cutoff phenomenon for reversible Markov chains were first systemically studied by C. and Saloff-Coste in [3]. Later in [1], C., Hsu and Sheu revealed more intrinsic mechanisms of `ℓ^2`-cutoffs and polished the cutoff criterion of C. and Saloff-Coste. Such a refinement makes some further theoretical analyses feasible including the comparison of `ℓ^2`-cutoffs between discrete time chains and continuous time chains.Keyword: Markov chains, reversibility, `ℓ^2`-distance
References
[1] Guan-Yu Chen, Jui-Ming Hsu, and Yuan-Chung Sheu. The `L^2`-cutoffs for reversible {M}arkov chains. Ann. Appl. Probab., 27(4):2305-2341, 2017.