We discuss the problem of finding the second largest eigenvalue of an operator
that defines a reversible Markov chain. The second largest eigenvalue governs
the rate at which the statistics of the Markov chain converge to equilibrium.
Scientific applications include understanding the very slow dynamics of some
models of dynamic glass. Applications in computing include estimating the rate
of convergence of Markov chain Monte Carlo algorithms.
Most practical Markov chains have state spaces so large that direct or even iterative methods from linear algebra are inapplicable. The size of the state space, which is the dimension of the eigenvalue problem, grows exponentially with the system size. This makes it impossible to store a vector (for sparse methods), let alone a matrix (for dense methods). Instead, we seek a method that uses only time correlation from samples produced from the Markov chain itself.
In this thesis, we propose a novel Krylov subspace type method to estimate the second eigenvalue from the simulation data of the Markov chain using test functions which are known to have good overlap with the slowest mode. This method starts with the naive Rayleigh quotient estimate of the test function and refines it to obtain an improved estimate of the second eigenvalue. We apply the method to a few model problems and the estimate compares very favorably with the known answer. We also apply the estimator to some Markov chains occuring in practice, most notably in the study of glasses. We show experimentally that our estimator is more accurate and stable for these problems compared to the existing methods.