A convergence framework for Airy${}_{\beta }$ line ensemble via pole evolution

Abstract

The Airy${}_{\beta }$ line ensemble is an infinite sequence of random curves. It is a natural extension of the Tracy-Widom${}_{\beta }$ distributions, and is expected to be the universal edge scaling limit of a range of models in random matrix theory and statistical mechanics. In this work, we provide a framework of proving convergence to the Airy${}_{\beta }$ line ensemble, via a characterization through the pole evolution of meromorphic functions satisfying certain stochastic differential equations. Our framework is then applied to prove the universality of the Airy${}_{\beta }$ line ensemble as the edge limit of various continuous time processes, including Dyson Brownian motions with general $\beta$ and potentials, Laguerre processes and Jacobi processes.

arXiv comment64 pages, 1 figure

References in corpus (39)

• Level-Spacing Distributions and the Airy Kernel
• Shape Fluctuations and Random Matrices
• On Orthogonal and Symplectic Matrix Ensembles
• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Matrix Models for Beta Ensembles
• Discrete polynuclear growth and determinantal processes
• Random matrices: Universality of local eigenvalue statistics up to the edge
• Fluctuation properties of the TASEP with periodic initial configuration
• Universality at the edge of the spectrum in Wigner random matrices
• Beta ensembles, stochastic Airy spectrum, and a diffusion
• Spectral Statistics of Erd{\H o}s-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues
• Spatial correlations of the 1D KPZ surface on a flat substrate
• Edge Universality of Beta Ensembles
• Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices
• From Random Matrices to Stochastic Operators
• Orthogonal polynomial ensembles in probability theory
• Global spectrum fluctuations for the $\beta$-Hermite and $\beta$-Laguerre ensembles via matrix models
• A Necessary and Sufficient Condition for Edge Universality of Wigner matrices
• Stochastic six-vertex model in a half-quadrant and half-line open ASEP
• Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field
• Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials
• Bulk properties of the Airy line ensemble
• The oriented swap process
• Zeros of Airy Function and Relaxation Process
• Non-Equilibrium Dynamics of Dyson's Model with an Infinite Number of Particles
• Airy processes with wanderers and new universality classes
• Optimal local law and central limit theorem for $\beta$-ensembles
• Universality of the Stochastic Airy Operator
• Edge rigidity and universality of random regular graphs of intermediate degree
• Infinite Dimensional Stochastic Differential Equations for Dyson's Model
• Error bounds for the large-argument asymptotic expansions of the Hankel and Bessel functions
• Edge scaling of the β-Jacobi ensemble
• A limit theorem at the spectral edge for corners of time-dependent Wigner matrices
• On the ubiquity of the Cauchy distribution in spectral problems
• KP hierarchy and trigonometric Calogero-Moser hierarchy
• Boundary current fluctuations for the half space ASEP and six vertex model
• Cutoff profile of the Metropolis biased card shuffling
• Pearcey universality at cusps of polygonal lozenge tiling
• A Limit Theorem for Stochastically Decaying Partitions at the Edge