Non-intersecting Brownian walkers and Yang-Mills theory on the sphere

Abstract

We study a system of N non-intersecting Brownian motions on a line segment [0,L] with periodic, absorbing and reflecting boundary conditions. We show that the normalized reunion probabilities of these Brownian motions in the three models can be mapped to the partition function of two-dimensional continuum Yang-Mills theory on a sphere respectively with gauge groups U(N), Sp(2N) and SO(2N). Consequently, we show that in each of these Brownian motion models, as one varies the system size L, a third order phase transition occurs at a critical value L=L_c(N)\sim \sqrt{N} in the large N limit. Close to the critical point, the reunion probability, properly centered and scaled, is identical to the Tracy-Widom distribution describing the probability distribution of the largest eigenvalue of a random matrix. For the periodic case we obtain the Tracy-Widom distribution corresponding to the GUE random matrices, while for the absorbing and reflecting cases we get the Tracy-Widom distribution corresponding to GOE random matrices. In the absorbing case, the reunion probability is also identified as the maximal height of N non-intersecting Brownian excursions ("watermelons" with a wall) whose distribution in the asymptotic scaling limit is then described by GOE Tracy-Widom law. In addition, large deviation formulas for the maximum height are also computed.

arXiv comment37 pages, 4 figures, revised and published version. A typo has been corrected in Eq. (10)

References in corpus (30)

• Level-Spacing Distributions and the Airy Kernel
• On Orthogonal and Symplectic Matrix Ensembles
• Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices
• Discrete polynuclear growth and determinantal processes
• Large $N$ Phase Transition in Continuum QCD${}_2$
• Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories
• Beta ensembles, stochastic Airy spectrum, and a diffusion
• Large Deviations of Extreme Eigenvalues of Random Matrices
• Two Dimensional QCD as a String Theory
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Large Deviations of the Maximum Eigenvalue in Wishart Random Matrices
• Instanton Induced Large $N$ Phase Transitions in Two and Four Dimensional QCD
• Exact distribution of the maximal height of p vicious walkers
• Vicious walkers, friendly walkers and Young tableaux II: With a wall
• Non-intersecting Brownian Interfaces and Wishart Random Matrices
• Nonintersecting Brownian excursions
• Extremal statistics of curved growing interfaces in 1+1 dimensions
• Brownian Motion, Chern-Simons Theory, and 2d Yang-Mills
• Classical Solutions for Two Dimensional QCD on the Sphere
• Maximum distributions of bridges of noncolliding Brownian paths
• Chern-Simons Theory, 2d Yang-Mills, and Lie Algebra Wanderers
• Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function
• One-dimensional stochastic growth and Gaussian ensembles of random matrices
• The height of watermelons with wall
• Evaluation of the Free Energy of Two-Dimensional Yang-Mills Theory
• Random walks and random fixed-point free involutions
• Universal Aspects of Two-Dimensional Yang-Mills Theory at Large N
• A new large N phase transition in YM2

Cited by in corpus (87)

• Growing interfaces uncover universal fluctuations behind scale invariance
• An exact solution for the KPZ equation with flat initial conditions
• Extreme value statistics of correlated random variables: a pedagogical review
• An appetizer to modern developments on the Kardar-Parisi-Zhang universality class
• Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence
• Top eigenvalue of a random matrix: large deviations and third order phase transition
• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
• Noninteracting fermions in a trap and random matrix theory
• How many eigenvalues of a Gaussian random matrix are positive?
• Extremes of N vicious walkers for large N: application to the directed polymer and KPZ interfaces
• Airy processes and variational problems
• Continuum statistics of the Airy2 process
• Local behavior and hitting probabilities of the Airy1 process
• Distribution of the time at which N vicious walkers reach their maximal height
• Exact extremal statistics in the classical $1d$ Coulomb gas
• A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix
• Dysonian dynamics of the Ginibre ensemble
• Reunion probability of N vicious walkers: typical and large fluctuations for large N
• Statistics of circular interface fluctuations in an off-lattice Eden model
• On the joint distribution of the maximum and its position of the Airy2 process minus a parabola
• Endpoint distribution of directed polymers in 1+1 dimensions
• Nonintersecting Brownian motions on the half-line and discrete Gaussian orthogonal polynomials
• Exact distributions of the number of distinct and common sites visited by N independent random walkers
• Extreme statistics and index distribution in the classical $1d$ Coulomb gas
• Universality of the third-order phase transition in the constrained Coulomb gas
• Unveiling the significance of eigenvectors in diffusing non-hermitian matrices by identifying the underlying Burgers dynamics
• Nonintersecting Brownian motions on the unit circle
• Universality of the weak pushed-to-pulled transition in systems with repulsive interactions
• Finite N corrections to the limiting distribution of the smallest eigenvalue of Wishart complex matrices
• Index Distribution of Cauchy Random Matrices
• Edge fluctuations and third-order phase transition in harmonically confined long-range systems
• Static fluctuations of a thick 1D interface in the 1+1 Directed Polymer formulation
• Non-intersecting Brownian bridges in the flat-to-flat geometry
• First-encounter time of two diffusing particles in confinement
• Transmutation of a Trans-series: The Gross-Witten-Wadia Phase Transition
• Interacting Particles on the Line and Dunkl Intertwining Operator of Type A: Application to the Freezing Regime
• From Painlevé to Zakharov-Shabat and beyond: Fredholm determinants and integro-differential hierarchies
• Periodic Airy process and equilibrium dynamics of edge fermions in a trap
• The height of watermelons with wall
• Large deviations of the top eigenvalue of large Cauchy random matrices
• Non-crossing Brownian paths and Dyson Brownian motion under a moving boundary
• Maximum relative height of elastic interfaces in random media
• Correlation Functions of XX0 Heisenberg Chain, q-Binomial Determinants, and Random Walks
• Free fermions and the classical compact groups
• Large-N ground state of the Lieb-Liniger model and Yang-Mills theory on a two-sphere
• Current fluctuations in stochastically resetting particle systems
• Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble
• The Heisenberg XX spin chain and low-energy QCD
• Third-order phase transition: random matrices and screened Coulomb gas with hard walls
• Stability of large complex systems with heterogeneous relaxation dynamics
• Maximum of an Airy process plus Brownian motion and memory in KPZ growth
• O'Connell's process as a vicious Brownian motion
• Convex Hulls of Multiple Random Walks: A Large-Deviation Study
• Finite-temperature and finite-time scaling of the directed polymer free-energy with respect to its geometrical fluctuations
• Static fluctuations of a thick 1D interface in the 1+1 Directed Polymer formulation: numerical study
• Noncolliding Brownian Motion with Drift and Time-Dependent Stieltjes-Wigert Determinantal Point Process
• Role of the Tracy-Widom distribution in the finite-size fluctuations of the critical temperature of the Sherrington-Kirkpatrick spin glass
• Hydrodynamic Limit of Multiple SLE
• Multiple phases and meromorphic deformations of unitary matrix models
• Free fermions and $\alpha$-determinantal processes
• The large-N limit for two-dimensional Yang-Mills theory
• Brownian Gibbs property for Airy line ensembles
• Critical exponents for higher order phase transitions: Landau theory and RG flow
• Brownian bridges for late time asymptotics of KPZ fluctuations in finite volume
• Tracy-Widom distribution as instanton sum of 2D IIA superstrings
• Instantons and Extreme Value Statistics of Random Matrices
• Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy--Widom GOE distribution
• Maximal distance travelled by N vicious walkers till their survival
• Mutually avoiding paths in random media and largests eigenvalues of random matrices
• KPZ fluctuations in finite volume
• Tail decay for the distribution of the endpoint of a directed polymer
• Ornstein-Uhlenbeck diffusion of hermitian and non-hermitian matrices - unexpected links
• Random ballistic growth and diffusion in symmetric spaces
• Large $N$ behaviour of the two-dimensional Yang-Mills partition function
• The phase diagram of $T\bar{T}$-deformed Yang-Mills theory on the sphere
• From elongated spanning trees to vicious random walks
• Constrained non-crossing Brownian motions, fermions and the Ferrari-Spohn distribution
• Discrete Toeplitz/Hankel determinants and the width of non-intersecting processes
• Painleve II in random matrix theory and related fields
• Scaling of many-particle correlations in a dissipative sandpile
• Heisenberg XX chain, non-homogeneously parameterised generating exponential, and diagonally restricted plane partitions
• Classical discrete symplectic ensembles on the linear and exponential lattice: skew orthogonal polynomials and correlation functions
• Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach
• Bessel process, Schramm-Loewner evolution, and Dyson model
• PhD thesis "Extreme value statistics of strongly correlated systems: fermions, random matrices and random walks"
• $T\bar{T}$ deformation of Calogero-Sutherland model via dimensional reduction
• Nonintersecting Brownian bridges between reflecting or absorbing walls