Abstract:
Characterising the average (and typical) ground-state energy (GSE) of mean-field disordered systems (or free energy at positive temperature) has been the focus of a large body of literature first in physics (e.g. via replica computations) and then rigorously in mathematics (e.g. via cavity, interpolation methods) in the past 40 years. This observable is generally expressed through a Parisi formula, i.e. an optimisation problem over a set of overlap probability distribution. In comparison, much less is known about the fluctuations of the GSE. In a recent series of work, I have derived an explicit expression for the large deviation function (LDF), characterising atypical fluctuations of the GSE, for spherical and Ising spin-glasses as well as for the elastic manifold. In this talk I will focus on the former. Interestingly, the expression for the LDF is again an optimisation problem over a skewed probability distribution function, allowing to characterise which configurations lead to an atypical fluctuation. The LDF generically displays a rich phase diagram with different patterns of replica-symmetry breaking. Most interestingly, I will show that the behaviour of the LDF in the vicinity of the average and typical GSE is universal and allows 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 (J. Stat. Phys. 191 (2), 11 (2024) / arXiv preprint arXiv:2306.11927) in collaboration with Y.V. Fyodorov (King’s College London) and P. Le Doussal (LPENS).