Painlevé Functions in Statistical Physics

2009/12/31 by Craig A. Tracy, Harold Widom · 2 citations
Mathematics · Physics and Astronomy · #Asymmetric simple exclusion process #Calculus (dental) #Epistemology #Lattice (music) #Limit (mathematics) #Limit of a function #Mathematical analysis #Mathematics #Philosophy #Physics #Random Matrices and Applications #Simple (philosophy) #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR #msc:34M55 #msc:60K35 #msc:82B23 #nlin.SI

paper · pdf · doi:10.2977/prims/38

published as Publ. RIMS Kyoto Univ., 47 (2011), 361-374 · Revised version updates some references

arxiv created 2010/04/27 · openalex publication_date 2011/03/24 · arxiv updated 2011/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We review recent progress in limit laws for the one-dimensional asymmetric simple exclusion process (ASEP) on the integer lattice. The limit laws are expressed in terms of a certain Painlevé II function. Furthermore, we take this opportunity to give a brief survey of the appearance of Painlevé functions in statistical physics.

Citations

• Quantum Inverse Scattering Method and Correlation Functions
• Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region
• A Fredholm Determinant Representation in ASEP
• Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices
• Fredholm determinants, differential equations and matrix models
• Limiting distributions for a polynuclear growth model with external sources
• Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals
• DIFFERENTIAL EQUATIONS FOR QUANTUM CORRELATION FUNCTIONS

Cited by

• Convergence from divergence
• Painlevé transcendents [wikipedia]

Related

• Painleve formulas of the limiting distributions for non-null complex sample covariance matrices
• Asymptotic properties of a family of solutions of the Painleve equation PVI
• Discrete Painlevé Equations
• On Painleve Related Functions Arising in Random Matrix Theory
• Application of the τ-function theory of Painlevé equations to random matrices: \PVI, the JUE, CyUE, cJUE and scaled limits