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Metastability and low lying spectra in reversible Markov chains. (English) Zbl 1010.60088
The authors continue their investigation of metastability [Probab. Theory Relat. Fields 119, No. 1, 99-161 (2001)]. In the present paper they study the connection between metastability and spectral theory in a more general setting. They consider a reversible Markov chain with transition matrix $$P_N$$ with a finite state space $$\Gamma_N$$, whose cardinality tends to infinity as $$N\to\infty$$. They define a set of metastable points each of them representing a metastable state. This notion is based on the idea that the time it takes to visit this set from anywhere is much smaller compared to the transition times between different metastable points. The set depends on the corresponding timescales. Under an additional non-degeneracy assumption the authors establish a precise relation between the low-lying eigenvalues of the operator $$1-P_N$$ and the mean metastable transition times with sharp estimates of the convergence of the distribution of these times to the exponential distribution.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C05 Classical dynamic and nonequilibrium statistical mechanics (general) 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
##### Keywords:
reversible Markov chains; metastability; low-lying spectrum
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