Hamilton–Jacobi equations for mean-field disordered systems

Mourrat, Jean-Christophe

Annales Henri Lebesgue, Tome 4 (2021), pp. 453-484.

Nous soutenons que les équations de Hamilton–Jacobi fournissent une approche pratique et intuitive pour étudier le comportement à grande échelle des systèmes désordonnés en champ moyen. Ce point de vue est illustré sur le problème de l’inférence d’une matrice de rang 1. Nous calculons la limite à grande échelle de l’énergie libre en montrant qu’elle satisfait une équation de Hamilton–Jacobi approximative avec un paramètre de viscosité et un terme d’erreur qui tendent vers zéro.

We argue that Hamilton–Jacobi equations provide a convenient and intuitive approach for studying the large-scale behavior of mean-field disordered systems. This point of view is illustrated on the problem of inference of a rank-one matrix. We compute the large-scale limit of the free energy by showing that it satisfies an approximate Hamilton–Jacobi equation with asymptotically vanishing viscosity parameter and error term.

Classification : 82B44, 82D30
Mots clés : spin glass, statistical inference, Hamilton–Jacobi equation

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