Metastability and Low Lying Spectra¶in Reversible Markov Chains |
| |
Authors: | Anton Bovier Michael Eckhoff Véronique Gayrard Markus Klein |
| |
Institution: | 1.Weierstrass-Institut für Angewandte Analysis und Stochastik, Mohrenstrasse 39, 10117 Berlin, Germany. E-mail: bovier@wias-berlin.de,DE;2.Institut für Mathematik, Universit?t Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany.?E-mail: meckhoff@math.uni-potsdam.de; mklein@math.uni-potsdam.de,DE;3.DMA, EPFL, 1021 Lausanne, Switzerland,CH |
| |
Abstract: | We study a large class of reversible Markov chains with discrete state space and transition matrix P
N
. We define the notion of a set of metastable points as a subset of the state space Γ
N
such that (i) this set is reached from any point x∈Γ
N
without return to x with probability at least b
N
, while (ii) for any two points x, y in the metastable set, the probability T
− 1
x
,
y
to reach y from x without return to x is smaller than a
N
− 1< b
N
. 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 −P
N
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.
Received: 1 August 2000 / Accepted: 19 November 2001 |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|