# M2 Project 2026-2027 ## Probabilistic Analysis in Mathematical Physics **Instructor**: Taro Kimura (office: A332) Here is a project proposal for *Probabilistic Analysis in Mathematical Physics* for M2 program of Math4Phys. ### Random matrix theory and related topics *Random Matrix Theory (RMT)* studies the statistical properties of eigenvalues of large random matrices and has become a central topic at the intersection of probability, mathematical physics, and statistical mechanics. One of the most remarkable discoveries in the field is the emergence of universal statistical laws that appear in a wide variety of seemingly unrelated systems. This project aims to explore random matrix theory as a gateway to modern developments in *integrable probability*. Particular emphasis will be placed on *determinantal point processes*, which provide an algebraic framework for analyzing eigenvalue statistics, and on the *Tracy--Widom distribution*, which describes the fluctuations of the largest eigenvalue in random matrix ensembles. These ideas have profound connections with growth processes and interacting particle systems belonging to the *Kardar--Parisi--Zhang (KPZ) universality class*. Through the study of classical random matrix ensembles and their asymptotic behavior, the project will illustrate how techniques from integrable probability lead to exact formulas and universal scaling limits. The ultimate goal is to understand the role of random matrices in the broader context of universal fluctuation phenomena and their connections to the KPZ equation and related stochastic models. #### Key words - Random matrix theory - Integrable probability - Determinantal point process - Tracy--Wisom distribution - Kardar--Parisi--Zhang (KPZ) equation #### References - Eynard, Kimura, Ribault, *Random Matrices*, https://arxiv.org/abs/1510.04430 - Baik, Daift, Suidan, *Combinatorics and Random Matrix Theory*, https://doi.org/10.1090/gsm/172