Universality for mathematical and physical systems

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.

New version contains some additional explication of the problems considered in the text and additional references

References in corpus (12)

• Random-Matrix Theory of Quantum Transport
• Level-Spacing Distributions and the Airy Kernel
• Alternating sign matrices and domino tilings
• The shape of a typical boxed plane partition
• Random Domino Tilings and the Arctic Circle Theorem
• Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices
• On the Distributions of the Lengths of the Longest Monotone Subsequences in Random Words
• A simple invariance theorem
• Analysis of airplane boarding via space-time geometry and random matrix theory
• Modelling gap-size distribution of parked cars using random-matrix theory
• One-dimensional stochastic growth and Gaussian ensembles of random matrices
• Growth models, random matrices and Painleve transcendents

Cited by in corpus (24)

• Random matrices: Universality of local eigenvalue statistics up to the edge
• Fluctuation properties of the TASEP with periodic initial configuration
• High Dimensional Statistical Inference and Random Matrices
• The Wasteland of Random Supergravities
• On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review
• Random matrices: Universality of local eigenvalue statistics
• Determinantal point processes and fermions on complex manifolds: Bulk universality
• The universal Airy_1 and Airy_2 processes in the Totally Asymmetric Simple Exclusion Process
• Anisotropic KPZ growth in 2+1 dimensions: fluctuations and covariance structure
• The shape of shortest paths in random spatial networks
• Poles of Painlevé IV Rationals and their Distribution
• Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model. Some history and some recent results
• Random matrices: The Four Moment Theorem for Wigner ensembles
• A geometrical triumvirate of real random matrices
• PDEs satisfied by extreme eigenvalues distributions of GUE and LUE
• Permutations avoiding 312 and another pattern, Chebyshev polynomials and longest increasing subsequences
• Empirical scaling of the length of the longest increasing subsequences of random walks
• Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GUE Revisited
• A Tracy-Widom Empirical Estimator For Valid P-values With High-Dimensional Datasets
• On Painleve Related Functions Arising in Random Matrix Theory
• Finite n Largest Eigenvalue Probability Distribution Function of Gaussian Ensembles
• Universality of fixation probabilities in randomly structured populations
• From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices
• Bulk asymptotics for polyanalytic correlation kernels