Spectral Analysis of Discrete Metastable Diffusions

We consider a discrete Schr\"odinger operator $H_\varepsilon= -\varepsilon^2\Delta_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon \mathbb Z^d)$, where $\varepsilon>0$ is a small parameter and the potential $V_\varepsilon$ is defined in terms of a multiwell energy landscape $f$ on $\mathbb R^d$. This operator can be seen as a discrete analog of the semiclassical Witten Laplacian of $\mathbb R^d$. It is unitarily equivalent to the generator of a diffusion on $\varepsilon \mathbb Z^d$, satisfying the detailed balance condition with respect to the Boltzmann weight $\exp{(-f/\varepsilon)}$. These type of diffusions exhibit metastable behaviour and arise in the context of disordered mean field models in Statistical Mechanics. We analyze the bottom of the spectrum of $H_\varepsilon$ in the semiclassical regime $\varepsilon\ll1$ 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.