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This lecture from the Harvard CMSA program on Classical, quantum, and probabilistic integrable systems explores the application of integrable structure in discrete random matrix theory. Discover how Roger Van Peski from Columbia University examines the emerging connections between integrability and discrete random matrices across integers, p-adic integers, and finite fields. Learn about key probabilistic results, including the convergence of discrete random matrix local limits to a new integrable interacting particle system called the 'reflecting Poisson sea.' Understand the exact formulas with Hall-Littlewood polynomials that enable these results, and explore intriguing newer formulas for Hermitian and antisymmetric p-adic matrices featuring 'formal' Hall-Littlewood processes with negative t parameter. The presentation is part of the novel interactions and applications series held on May 15, 2025.
