Integrable Probability and Universality in Mathematical Physics and Machine Learning

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This proposal focuses on the study of universal behaviors in critical phenomena at the interface between some probabilistic models (random matrices, interacting particle systems) and the theory of integrable systems (e.g. nonlinear wave equations). Integrable systems consist in a class of overdetermined sets of partial differential equations whose solutions can be thoroughly analyzed via a suitable nonlinear transformation (Scattering Transform) that reduces the complicated, nonlinear dynamics into a linear one. In certain critical (e.g. phase transition) or asymptotic (e.g. large-time, continuum limit) regimes, Integrable System models or Random Matrix ensembles display remarkable universality patterns, such that their behaviour become independent on the initial data (even in the presence of randomness) or on the specific details of the probability distribution they are based on. The motivations for this work derive from a plethora of models of current physical interest, including crystal and polymer growth phenomena, mutually-avoiding multiple random-walkers, nonlinear phenomena in optics, polymers, superconductors (e.g. Bose-Einstein condensate) and fluids (most notably, rogue waves). This theoretical research will additionally be applied to understand Machine Learning (ML) phenomena and models, by describing them through a much needed rigorous framework. The main strategy for all these instances will be their reformulation in terms of a particular boundary value problem, the so-called Riemann-Hilbert problem (RHP) and its geometrical connection with Painlevé equations and isomonodromic tau functions. The research agenda is organized into the following parallel directions of research: (1) analyze universality properties of statistical quantities of determinantal point processes and random matrix models within certain asymptotic regimes, and their geometrical interpretation; (2) analyze asymptotic behaviour of dispersive integrable PDEs (e.g. Korteweg-de Vries, Nonlinear Schrödinger equations) with randomness: in particular, solitons and soliton gasses; (3) study universality aspects of overparametrized ML models: investigate the emerging of integrable structures in certain asymptotic or critical regimes (e.g. long time training, infinitely deep/wide networks, mean-field regime, neural tangent kernel regime, etc.) and describe the generalization risk curve of deep models.