Spectral Analysis of Discrete Metastable Diffusions

Abstract

AbstractWe consider a discrete Schrödinger operator $$ H_\varepsilon = -\varepsilon ^2\Delta _\varepsilon + V_\varepsilon $$ H ε = - ε 2 Δ ε + V ε on $$\ell ^2(\varepsilon \mathbb {Z}^d)$$ ℓ 2 ( ε Z d ) , where $$\varepsilon >0$$ ε > 0 is a small parameter and the potential $$V_\varepsilon $$ V ε is defined in terms of a multiwell energy landscape f on $$\mathbb {R}^d$$ R d . This operator can be seen as a discrete analog of the semiclassical Witten Laplacian of $$\mathbb {R}^d$$ R d . It is unitarily equivalent to the generator of a diffusion on $$\varepsilon \mathbb {Z}^d$$ ε Z d , satisfying the detailed balance condition with respect to the Boltzmann weight $$\exp {(-f/\varepsilon )}$$ 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 $$H_\varepsilon $$ H ε in the semiclassical regime $$\varepsilon \ll 1$$ 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.