Bertrand Lacroix-A-Chez-Toine

Fluctuations of the ground-state energy of spherical spin-glasses

Disordered systems, such as a spin glasses, are defined from a set of high-dimensional configurations, the degrees of freedom, and a Hamiltonian, which prescribes a random energy to each configuration, resulting from the complex interactions between these. A natural question for this type of system is to find the ground-state energy (GSE), i.e. the lowest energy accessible by any configuration of the system. In the high-dimensional limit, this GSE is self-averaging and for almost every realisation, the GSE coincides with its average. The latter is expressed as the optimum of a nontrivial functional optimisation problem known as Parisi formula. The rare fluctuations of the GSE away from its average have, in comparison, received much less attention. I will show how non-rigorous “replica computations” can be used to derive the large deviation function at speed $N$ of the GSE for spherical spin glasses in terms of a functional optimisation process, extending Parisi’s formula. This large-deviation function generically displays a rich phase diagram. Most interestingly, I will show that the behaviour of the large deviation function in the vicinity of the average and typical GSE is universal and can be used to make precise predictions on the non-trivial tails of the distribution of typical fluctuations of the GSE.

This talk is based on an article in collaboration with Yan V. Fyodorov and Pierre Le Doussal (J. Stat. Phys. 191 (2), 11 (2024) / arXiv preprint arXiv:2306.11927).