Session S38 - Geometric Potential Analysis

Friday, July 16, 16:40 ~ 17:10 UTC-3

Discrete Witten Laplacians and metastability of disordered mean field models

Giacomo Di Gesù

University of Pisa , Italy   -   giacomo.digesu@unipi.it

We consider a discrete Schrödinger operator $H_\varepsilon= -\varepsilon^2\Delta_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon \mathbb Z^d)$, where $V_\varepsilon$ is defined in terms of a general multiwell energy landscape $f$ on $\mathbb R^d$. This operator can be seen as a discrete analog of the semiclassical Witten Laplacian of $\mathbb R^d$. Moreover it is unitarily equivalent to a type of discrete diffusion arising in the context of disordered mean field models in Statistical Mechanics, as e.g. the Curie-Weiss model.

In this talk I will present results on the bottom of the spectrum of $H_\varepsilon$ in the semiclassical regime $\varepsilon\ll1$, including the fine asymptotics of the tunnel effect between wells. These results require minimal regularity assumptions on f, are based on microlocalization techniques and permit to recover the Eyring-Kramers formula for the metastable tunneling time of the underlying stochastic process.

Further I will discuss the complex property and the Hodge-type extension of the discrete Witten Laplacian to the full algebra of discrete differential forms. This is inspired by the well-known fact that lifting the continuous space Witten Laplacian to higher forms provides a powerful tool for studying e.g. Morse inequalities and functional inequalities (Brascamp-Lieb, Bakry-Émery, Helffer-Sjöstrand) on manifolds.

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