Browsing by Author "Kumagai, Takashi"

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Cutoffs for product chains
(2018-11-01)
Chen, Guan-Yu
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Kumagai, Takashi
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應用數學系
;
Department of Applied Mathematics
We consider products of ergodic Markov chains and discuss their cutoffs in total variation. Our framework is general in that rates to pick up coordinates are not necessary equal, and different coordinates may correspond to distinct chains. We give necessary and sufficient conditions for cutoffs of product chains in terms of those of coordinate chains under certain conditions. A comparison of mixing times between the product chain and its coordinate chains is made in detail as well. Examples are given to show that neither cutoffs for product chains nor for coordinate chains imply others in general. (C) 2018 Elsevier B.V. All rights reserved.
PRODUCTS OF RANDOM WALKS ON FINITE GROUPS WITH MODERATE GROWTH
(2019-01-01)
Chen, Guan-Yu
;
Kumagai, Takashi
;
應用數學系
;
Department of Applied Mathematics
In this article, we consider products of random walks on finite groups with moderate growth and discuss their cutoffs in the total variation. Based on several comparison techniques, we are able to identify the total variation cutoff of discrete time lazy random walks with the Hellinger distance cutoff of continuous time random walks. Along with the cutoff criterion for Laplace transforms, we derive a series of equivalent conditions on the existence of cutoffs, including the existence of pre-cutoffs, Peres' product condition and a formula generated by the graph diameters. For illustration, we consider products of Heisenberg groups and randomized products of finite cycles.