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Seminar 217, Risk Management: Asymptotic Spectral Analysis of Markov Chains with Rare Transitions: A GraphAlgorithmic ApproachSeminar: Risk Seminar  October 16  11 a.m.12:30 p.m.  1011 Evans Hall Speaker: Tingyue Gan, UC Berkeley Consortium for Data Analytics in Risk Parameterdependent Markov chains with exponentially small transition rates arise in modeling complex systems in physics, chemistry, and biology. Such processes often manifest metastability, and the spectral properties of the generators largely govern their longterm dynamics. In this work, we propose a constructive graphalgorithmic approach to computing the asymptotic estimates of eigenvalues and eigenvectors of the generator. In particular, we introduce the concepts of the hierarchy of Typical Transition Graphs and the associated sequence of Characteristic Timescales. Typical Transition Graphs can be viewed as a unification of Wentzell’s hierarchy of optimal Wgraphs and Friedlin’s hierarchy of Markov chains, and they are capable of describing typical escapes from metastable classes as well as cyclic behaviors within metastable classes, for both reversible and irreversible processes. We apply the proposed approach to conduct zerotemperature asymptotic analysis of the stochastic network representing the energy landscape of the LennardJones cluster of 75 atoms. 

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