## View abstract

### 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   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloaka3a7dcda9d080a1869b5a6042af051dd').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addya3a7dcda9d080a1869b5a6042af051dd = 'g&#105;&#97;c&#111;m&#111;.d&#105;g&#101;s&#117;' + '&#64;'; addya3a7dcda9d080a1869b5a6042af051dd = addya3a7dcda9d080a1869b5a6042af051dd + '&#117;n&#105;p&#105;' + '&#46;' + '&#105;t'; var addy_texta3a7dcda9d080a1869b5a6042af051dd = 'g&#105;&#97;c&#111;m&#111;.d&#105;g&#101;s&#117;' + '&#64;' + '&#117;n&#105;p&#105;' + '&#46;' + '&#105;t';document.getElementById('cloaka3a7dcda9d080a1869b5a6042af051dd').innerHTML += '<a ' + path + '\'' + prefix + ':' + addya3a7dcda9d080a1869b5a6042af051dd + '\'>'+addy_texta3a7dcda9d080a1869b5a6042af051dd+'<\/a>';

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.