The theory of L2-spectral gaps for reversible Markov chains has been studied by many authors. In this paper we consider positive recurrent general state space Markov chains with stationary transition probabilities. Replacing the assumption of reversibility with a weaker assumption, we still obtain a simple necessary and sufficient condition for the spectral gap property of the associated Markov operator in terms of the isoperimetric constant. We show that this result can be applied to a large class of Markov chains, including those that are related to positive recurrent finite-range random walks on Z.

In this paper we study the eigenvalues and eigenfunctions of metric measure manifolds. We prove that any eigenfunction is $C^{1,\alpha}$ at its critical points and $C^{\infty}$ elsewhere. Moreover, the eigenfunction corresponding to the first eigenvalue in the Dirichlet problem does not change sign. We also discuss the first eigenvalue, the Sobolev constants and their relationship with the isoperimetric constants. 2000 Mathematics Subject Classification: 47J05, 47J10, 53C60, 58E05, 58C40.

This article describes new estimates for the second largest eigenvalue in absolute value of reversible and ergodic Markov chains on finite state spaces. These estimates apply when the stationary distribution assigns a probability higher than 0.702 to some given state of the chain. Geometric tools are used. The bounds mainly involve the isoperimetric constant of the chain, and hence generalize famous results obtained for the second eigenvalue. Comparison estimates are also established, using the isoperimetric constant of a reference chain. These results apply to the Metropolis-Hastings algorithm in order to solve minimization problems, when the probability of obtaining the solution from the algorithm can be chosen beforehand. For these dynamics, robust bounds are obtained at moderate levels of concentration.