Distribution of the time at which N vicious walkers reach their maximal height
arXiv:1102.1640 · doi:10.1103/PhysRevE.83.061146
Abstract
We study the extreme statistics of N non-intersecting Brownian motions (vicious walkers) over a unit time interval in one dimension. Using path-integral techniques we compute exactly the joint distribution of the maximum M and of the time τ_M at which this maximum is reached. We focus in particular on non-intersecting Brownian bridges ("watermelons without wall") and non-intersecting Brownian excursions ("watermelons with a wall"). We discuss in detail the relationships between such vicious walkers models in watermelons configurations and stochastic growth models in curved geometry on the one hand and the directed polymer in a disordered medium (DPRM) with one free end-point on the other hand. We also check our results using numerical simulations of Dyson's Brownian motion and confront them with numerical simulations of the Polynuclear Growth Model (PNG) and of a model of DPRM on a discrete lattice. Some of the results presented here were announced in a recent letter [J. Rambeau and G. Schehr, Europhys. Lett. 91, 60006 (2010)].
arXiv comment30 pages, 12 figures
References in corpus (46)
- 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
- Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices
- The one-dimensional KPZ equation: an exact solution and its universality
- Discrete polynuclear growth and determinantal processes
- Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals
- Universality Classes for Extreme Value Statistics
- Free-energy distribution of the directed polymer at high temperature
- Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers
- Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and SinhGordon models
- Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces
- Exact Maximal Height Distribution of Fluctuating Interfaces
- Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
- Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
- Generalised extreme value statistics and sum of correlated variables
- Random Convex Hulls and Extreme Value Statistics
- Maximal Height Scaling of Kinetically Growing Surfaces
- Exact distribution of the maximal height of p vicious walkers
- Statistical Mechanics of Logarithmic REM: Duality, Freezing and Extreme Value Statistics of Noises generated by Gaussian Free Fields
- Extreme Value Statistics of Hierarchically Correlated Variables: Deviation from Gumbel Statistics and Anomalous Persistence
- Convex Hull of N Planar Brownian Motions: Exact Results and an Application to Ecology
- Probability distribution of the free energy of a directed polymer in a random medium
- Exact solutions for the statistics of extrema of some random 1D landscapes, Application to the equilibrium and the dynamics of the toy model
- On the time to reach maximum for a variety of constrained Brownian motions
- Non-intersecting Brownian Interfaces and Wishart Random Matrices
- Universal Asymptotic Statistics of Maximal Relative Height in One-dimensional Solid-on-solid Models
- Maximal height statistics for 1/f^alpha signals
- Extreme value statistics from the Real Space Renormalization Group: Brownian Motion, Bessel Processes and Continuous Time Random Walks
- Time to reach the maximum for a random acceleration process
- An Anisotropic Ballistic Deposition Model with Links to the Ulam Problem and the Tracy-Widom Distribution
- Extremal statistics of curved growing interfaces in 1+1 dimensions
- Statistics of extremal intensities for Gaussian interfaces
- Extreme statistics for time series: Distribution of the maximum relative to the initial value
- Maximum distributions of bridges of noncolliding Brownian paths
- Ground state energy of a non-integer number of particles with delta attractive interactions
- Maximum of N Independent Brownian Walkers till the First Exit From the Half Space
- Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function
- Ecological Invasion, Roughened Fronts, and a Competitor's Extreme Advance: Integrating Stochastic Spatial-Growth Models
- Vicious Walkers in a Potential
- Temperature-induced crossovers in the static roughness of a one-dimensional interface
- Fluctuations of the winding number of a directed polymer in a random medium
- Maximum relative height of one-dimensional interfaces : from Rayleigh to Airy distribution
- Sylvester's question and the Random Acceleration Process
- The height and range of watermelons without wall
Cited by in corpus (35)
- Growing interfaces uncover universal fluctuations behind scale invariance
- Extreme value statistics of correlated random variables: a pedagogical review
- 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
- Extremes of N vicious walkers for large N: application to the directed polymer and KPZ interfaces
- Airy processes and variational problems
- Generalised `Arcsine' laws for run-and-tumble particle in one dimension
- Local behavior and hitting probabilities of the Airy1 process
- Continuum statistics of the Airy2 process
- 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
- Distribution of the time of the maximum for stationary processes
- Nonintersecting Brownian motions on the half-line and discrete Gaussian orthogonal polynomials
- Time between the maximum and the minimum of a stochastic process
- Distribution of the Time Between Maximum and Minimum of Random Walks
- Invariant -Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law
- Extreme value statistics and arcsine laws for heterogeneous diffusion processes
- Time to reach the maximum for a stationary stochastic process
- Non-intersecting Brownian bridges in the flat-to-flat geometry
- Periodic Airy process and equilibrium dynamics of edge fermions in a trap
- Maximum relative height of elastic interfaces in random media
- Non-crossing Brownian paths and Dyson Brownian motion under a moving boundary
- Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble
- Extremal statistics of a one dimensional run and tumble particle with an absorbing wall
- Stability of large complex systems with heterogeneous relaxation dynamics
- O'Connell's process as a vicious Brownian motion
- Convex Hulls of Multiple Random Walks: A Large-Deviation Study
- Extremal statistics for a resetting Brownian motion before its first-passage time
- Brownian Gibbs property for Airy line ensembles
- Maximal distance travelled by N vicious walkers till their survival
- Generalized arcsine laws for a sluggish random walker with subdiffusive growth
- Determinantal process starting from an orthogonal symmetry is a Pfaffian process
- Extremal statistics for a one-dimensional Brownian motion with a reflective boundary
- Exact joint distributions of three global characteristic times for Brownian motion
- A stationary model of non-intersecting directed polymers