Exact distribution of the maximal height of p vicious walkers

Abstract

Using path integral techniques, we compute exactly the distribution of the maximal height H_p of p nonintersecting Brownian walkers over a unit time interval in one dimension, both for excursions (p-watermelons with a wall) and bridges (p-watermelons without a wall), for all integer p\ge 1. For large p, we show that < H_p > \sim \sqrt{2p} (excursions) whereas < H_p > \sim \sqrt{p} (bridges). Our exact results prove that previous numerical experiments only measured the pre-asymptotic behaviors and not the correct asymptotic ones. In addition, our method establishes a physical connection between vicious walkers and random matrix theory.

arXiv comment4 pages, 2 figures; minor modifications, added references; published version

References in corpus (14)

• Level-Spacing Distributions and the Airy Kernel
• Shape Fluctuations and Random Matrices
• On Orthogonal and Symplectic Matrix Ensembles
• Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces
• Exact Maximal Height Distribution of Fluctuating Interfaces
• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Vicious walkers, friendly walkers and Young tableaux II: With a wall
• Universal Asymptotic Statistics of Maximal Relative Height in One-dimensional Solid-on-solid Models
• Maximal height statistics for 1/f^alpha signals
• Nonintersecting Brownian excursions
• Extreme statistics for time series: Distribution of the maximum relative to the initial value
• Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function
• Vicious Walkers in a Potential
• Beyond the Wigner Distribution: Schr{ö}dinger Equations and Terrace Width Distributions

Cited by in corpus (73)

• Extreme value statistics of correlated random variables: a pedagogical review
• Top eigenvalue of a random matrix: large deviations and third order phase transition
• Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• Random Convex Hulls and Extreme Value Statistics
• Non-intersecting Brownian Interfaces and Wishart Random Matrices
• 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
• Statistical mechanics of complex neural systems and high dimensional data
• Airy processes and variational problems
• Extreme value statistics from the Real Space Renormalization Group: Brownian Motion, Bessel Processes and Continuous Time Random Walks
• Distribution of the time at which N vicious walkers reach their maximal height
• Extremal statistics of curved growing interfaces in 1+1 dimensions
• Extremal statistics for stochastic resetting systems
• 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
• 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
• Invariant $\beta$-Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law
• Maximum distributions of bridges of noncolliding Brownian paths
• Harmonically confined particles with long-range repulsive interactions
• Extreme value statistics and arcsine laws for heterogeneous diffusion processes
• Unveiling the significance of eigenvectors in diffusing non-hermitian matrices by identifying the underlying Burgers dynamics
• Geometrical optics of constrained Brownian excursion: from the KPZ scaling to dynamical phase transitions
• Maximum of N Independent Brownian Walkers till the First Exit From the Half Space
• Nonintersecting Brownian motions on the unit circle
• Finite N corrections to the limiting distribution of the smallest eigenvalue of Wishart complex matrices
• Index Distribution of Cauchy Random Matrices
• On certain functionals of the maximum of Brownian motion and their applications
• Unfolding Tagged Particle Histories in Single-File Diffusion: Exact Single- and Two-Tag Local Times Beyond Large Deviation Theory
• Non-intersecting Brownian bridges in the flat-to-flat geometry
• Periodic Airy process and equilibrium dynamics of edge fermions in a trap
• Anomalous 1D fluctuations of a simple 2D random walk in a large deviation regime
• Non-crossing Brownian paths and Dyson Brownian motion under a moving boundary
• The height of watermelons with wall
• Multiplicative functionals on ensembles of non-intersecting paths
• Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble
• Bose--Einstein Condensation in the Large Deviations Regime with Applications to Information System Models
• Extremal statistics of a one dimensional run and tumble particle with an absorbing wall
• Multiple phases and vicious walkers in a wedge
• Maximum of an Airy process plus Brownian motion and memory in KPZ growth
• Dynamical phase transition in the occupation fraction statistics for non-crossing Brownian particles
• O'Connell's process as a vicious Brownian motion
• Convex Hulls of Random Walks in Higher Dimensions: A Large Deviation Study
• Noncolliding Brownian Motion with Drift and Time-Dependent Stieltjes-Wigert Determinantal Point Process
• Convex Hulls of Multiple Random Walks: A Large-Deviation Study
• Optimal potentials for diffusive search strategies
• Capture of a diffusing lamb by a diffusing lion when both return home
• Extreme Value Statistics and Arcsine Laws of Brownian Motion in the Presence of a Permeable Barrier
• Vicious Lévy flights
• Brownian bridges for late time asymptotics of KPZ fluctuations in finite volume
• Universal shocks in the Wishart random-matrix ensemble - a sequel
• Slow Kinetics of Brownian Maxima
• Persistence of Random Walk Records
• The Ising Limit of the XXZ Heisenberg Magnet and Certain Thermal Correlation Functions
• Maximal distance travelled by N vicious walkers till their survival
• Mutually avoiding paths in random media and largests eigenvalues of random matrices
• Determinantal process starting from an orthogonal symmetry is a Pfaffian process
• Fluctuations of ring polymers
• Ornstein-Uhlenbeck diffusion of hermitian and non-hermitian matrices - unexpected links
• Reciprocal Time Relation of Noncolliding Brownian Motion with Drift
• Stretching of a fractal polymer around a disc reveals KPZ-like statistics
• 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
• Area Distribution of Elastic Brownian Motion
• Scaling of many-particle correlations in a dissipative sandpile
• Characteristic Polynomials of Random Matrices and Noncolliding Diffusion Processes
• PhD thesis "Extreme value statistics of strongly correlated systems: fermions, random matrices and random walks"
• Extreme value distributions of noncolliding diffusion processes