Metastability and Low Lying Spectra¶in Reversible Markov Chains

By Anton Bovier, M. Eckhoff, V. Gayrard, M. Klein. 2000

254 citations recorded by the source corpus.

Abstract: We study a large class of reversible Markov chains with discrete state space and transition matrix PN. We define the notion of a set of metastable points as a subset of the state space ΓN such that (i) this set is reached from any point x∈ΓN without return to x with probability at least bN, while (ii) for any two points x, y in the metastable set, the probability T− 1x,y to reach y from x without return to x is smaller than aN− 1< bN. Under some additional non-degeneracy assumption, we show that in such a situation:(i) To each metastable point corresponds a metastable state, whose mean exit time can be computed precisely.(ii) To each metastable point corresponds one simple eigenvalue of 1 −PN which is essentially equal to the inverse mean exit time from this state. Moreover, these results imply very sharp uniform control of the deviation of the probability distribution of metastable exit times from the exponential distribution.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c50d9026d43a17bb05c775ce8ba8711b73084d13

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Citations in the corpus

Metastability: A Potential-Theoretic Approach (2016, background): “It can, however, be improved to give useful bounds under tightness conditions on Q [5].”
Metastability and Dominant Eigenvalues of Transfer Operators (2006, background): “, exit times [3, 14, 15], eigenvalues of transfer operators or generators [3, 10, 9, 7, 27], phase transitions [2, 6], reduced Markovian approximations [27, 20, 21], averaging [28], and many other areas.”
Transition Path Theory for Markov Jump Processes (2009, unknown): “[8, 11] in the context of time-reversible chains.”
On the Approximation Quality of Markov State Models (2010, background): “Markov state models (MSMs) have been considered for processes that have metastable dynamics [1, 2, 3, 4], especially in molecular dynamics.”
Tunneling and Metastability of Continuous Time Markov Chains (2009, result): “In contrast with the potential approach proposed in [4, 6, 7 ], the definitions and the method we present here does not depend on a reversibility assumption, though it becomes particularly simple in this case.”
On Times to Quasi-stationarity for Birth and Death Processes (2009, background): “Miclo [21]). Furthermore, for those eigenvalues, the time Ti and τ (N ) i are equivalent and are supposedly also close to the exit times associated to certain cycles, which are known to be almost exponential variables with eigenvalues as parameters (see, for instance, Bovier, Eckhoff, Gayrard, and K”