Extremal statistics of curved growing interfaces in 1+1 dimensions

We study the joint probability distribution function (pdf) of the maximum M of the height and its position X_M of a curved growing interface belonging to the universality class described by the Kardar-Parisi-Zhang equation in 1+1 dimensions. We obtain exact results for the closely related problem of p non-intersecting Brownian bridges where we compute the joint pdf P_p(M,\tau_M) where \tau_M is there the time at which the maximal height M is reached. Our analytical results, in the limit p \to \infty, become exact for the interface problem in the growth regime. We show that our results, for moderate values of p \sim 10 describe accurately our numerical data of a prototype of these systems, the polynuclear growth model in droplet geometry. We also discuss applications of our results to the ground state configuration of the directed polymer in a random potential with one fixed endpoint.Comment: 6 pages, 4 figures. Published version, to appear in Europhysics Letters. New results added for non-intersecting excursion

Maximum relative height of one-dimensional interfaces : from Rayleigh to Airy distribution

We introduce an alternative definition of the relative height h^\kappa(x) of a one-dimensional fluctuating interface indexed by a continuously varying real paramater 0 \leq \kappa \leq 1. It interpolates between the height relative to the initial value (i.e. in x=0) when \kappa = 0 and the height relative to the spatially averaged height for \kappa = 1. We compute exactly the distribution P^\kappa(h_m,L) of the maximum h_m of these relative heights for systems of finite size L and periodic boundary conditions. One finds that it takes the scaling form P^\kappa(h_m,L) = L^{-1/2} f^\kappa (h_m L^{-1/2}) where the scaling function f^\kappa(x) interpolates between the Rayleigh distribution for \kappa=0 and the Airy distribution for \kappa=1, the latter being the probability distribution of the area under a Brownian excursion over the unit interval. For arbitrary \kappa, one finds that it is related to, albeit different from, the distribution of the area restricted to the interval [0, \kappa] under a Brownian excursion over the unit interval.Comment: 25 pages, 4 figure

Distribution of the time at which N vicious walkers reach their maximal height

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 \tau_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)].Comment: 30 pages, 12 figure

A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix

In this paper, we first briefly review some recent results on the distribution of the maximal eigenvalue of a $(N\times N)$ random matrix drawn from Gaussian ensembles. Next we focus on the Gaussian Unitary Ensemble (GUE) and by suitably adapting a method of orthogonal polynomials developed by Gross and Matytsin in the context of Yang-Mills theory in two dimensions, we provide a rather simple derivation of the Tracy-Widom law for GUE. Our derivation is based on the elementary asymptotic scaling analysis of a pair of coupled nonlinear recursion relations. As an added bonus, this method also allows us to compute the precise subleading terms describing the right large deviation tail of the maximal eigenvalue distribution. In the Yang-Mills language, these subleading terms correspond to non-perturbative (in $1/N$ expansion) corrections to the two-dimensional partition function in the so called `weak' coupling regime.Comment: 2 figure

Endpoint distribution of directed polymers in 1+1 dimensions

We give an explicit formula for the joint density of the max and argmax of the Airy$_2$ process minus a parabola. The argmax has a universal distribution which governs the rescaled endpoint for large time or temperature of directed polymers in 1+1 dimensions.Comment: Expanded introductio

Airy processes and variational problems

We review the Airy processes; their formulation and how they are conjectured to govern the large time, large distance spatial fluctuations of one dimensional random growth models. We also describe formulas which express the probabilities that they lie below a given curve as Fredholm determinants of certain boundary value operators, and the several applications of these formulas to variational problems involving Airy processes that arise in physical problems, as well as to their local behaviour.Comment: Minor corrections. 41 pages, 4 figures. To appear as chapter in "PASI Proceedings: Topics in percolative and disordered systems

Extreme value statistics from the Real Space Renormalization Group: Brownian Motion, Bessel Processes and Continuous Time Random Walks

We use the Real Space Renormalization Group (RSRG) method to study extreme value statistics for a variety of Brownian motions, free or constrained such as the Brownian bridge, excursion, meander and reflected bridge, recovering some standard results, and extending others. We apply the same method to compute the distribution of extrema of Bessel processes. We briefly show how the continuous time random walk (CTRW) corresponds to a non standard fixed point of the RSRG transformation.Comment: 24 pages, 5 figure

Nonintersecting Brownian motions on the half-line and discrete Gaussian orthogonal polynomials

We study the distribution of the maximal height of the outermost path in the model of $N$ nonintersecting Brownian motions on the half-line as $N\to \infty$, showing that it converges in the proper scaling to the Tracy-Widom distribution for the largest eigenvalue of the Gaussian orthogonal ensemble. This is as expected from the viewpoint that the maximal height of the outermost path converges to the maximum of the $\textrm{Airy}_2$ process minus a parabola. Our proof is based on Riemann-Hilbert analysis of a system of discrete orthogonal polynomials with a Gaussian weight in the double scaling limit as this system approaches saturation. We consequently compute the asymptotics of the free energy and the reproducing kernel of the corresponding discrete orthogonal polynomial ensemble in the critical scaling in which the density of particles approaches saturation. Both of these results can be viewed as dual to the case in which the mean density of eigenvalues in a random matrix model is vanishing at one point.Comment: 39 pages, 4 figures; The title has been changed from "The limiting distribution of the maximal height of nonintersecting Brownian excursions and discrete Gaussian orthogonal polynomials." This is a reflection of the fact that the analysis has been adapted to include nonintersecting Brownian motions with either reflecting of absorbing boundaries at zero. To appear in J. Stat. Phy

Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence

We provide a comprehensive report on scale-invariant fluctuations of growing interfaces in liquid-crystal turbulence, for which we recently found evidence that they belong to the Kardar-Parisi-Zhang (KPZ) universality class for 1+1 dimensions [Phys. Rev. Lett. 104, 230601 (2010); Sci. Rep. 1, 34 (2011)]. Here we investigate both circular and flat interfaces and report their statistics in detail. First we demonstrate that their fluctuations show not only the KPZ scaling exponents but beyond: they asymptotically share even the precise forms of the distribution function and the spatial correlation function in common with solvable models of the KPZ class, demonstrating also an intimate relation to random matrix theory. We then determine other statistical properties for which no exact theoretical predictions were made, in particular the temporal correlation function and the persistence probabilities. Experimental results on finite-time effects and extreme-value statistics are also presented. Throughout the paper, emphasis is put on how the universal statistical properties depend on the global geometry of the interfaces, i.e., whether the interfaces are circular or flat. We thereby corroborate the powerful yet geometry-dependent universality of the KPZ class, which governs growing interfaces driven out of equilibrium.Comment: 31 pages, 21 figures, 1 table; references updated (v2,v3); Fig.19 updated & minor changes in text (v3); final version (v4); J. Stat. Phys. Online First (2012