Universality for mathematical and physical systems
Percy Deift
TL;DR
The paper argues that universal behavior arising in equilibrium physics has deep parallels in a wide class of mathematical problems. By modeling diverse systems with random matrix theory across the ensembles $\beta=1,2,4$, it derives universal scaling limits: bulk correlations converge to the sine kernel $K_\infty$, while edge statistics converge to Airy-type kernels and associated Tracy–Widom distributions $F_\beta$. It synthesizes results from seven problems—ranging from neutron resonances to longest increasing subsequences and tiling models—showing how they reduce to RMT predictions (GOE/GUE/GSE) in appropriate limits, often via combinatorial correspondences such as Robinson–Schensted and Plancherel measure. The work also outlines the analytic machinery (Riemann–Hilbert methods and Painlevé equations) used to establish these universality results and ends by outlining avenues toward a broader, macroscopic mathematical framework for universal phenomena.
Abstract
All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all governed by the same laws and formulae of thermodynamics. In this paper we describe some recent history of universality ideas in physics starting with Wigner's model for the scattering of neutrons off large nuclei and show how these ideas have led mathematicians to investigate universal behavior for a variety of mathematical systems. This is true not only for systems which have a physical origin, but also for systems which arise in a purely mathematical context such as the Riemann hypothesis, and a version of the card game solitaire called patience sorting.