Spectral Analysis of Discrete Metastable Diffusions

We consider a discrete Schrödinger operator H ε = - ε 2 Δ ε + V ε on ℓ 2 ( ε Z d ) , where ε > 0 is a small parameter and the potential V ε is defined in terms of a multiwell energy landscape f on R d . This operator can be seen as a discrete analog of the semiclassical Witten Laplacian of R d . It is unitarily equivalent to the generator of a diffusion on ε Z d , satisfying the detailed balance condition with respect to the Boltzmann weight exp ( - f / ε ) . These type of diffusions exhibit metastable behavior and arise in the context of disordered mean field models in Statistical Mechanics. We analyze the bottom of the spectrum of H ε in the semiclassical regime ε ≪ 1 and show that there is a one-to-one correspondence between exponentially small eigenvalues and local minima of f . Then we analyze in more detail the bistable case and compute the precise asymptotic splitting between the two exponentially small eigenvalues. Through this purely spectral-theoretical analysis of the discrete Witten Laplacian we recover in a self-contained way the Eyring–Kramers formula for the metastable tunneling time of the underlying stochastic process.