The Height and Range of Watermelons without Wall

Abstract. -We study the joint probability distribution function (pdf) Pt(M, XM ) of the maximum M of the height and its position XM of a curved growing interface belonging to the universality class described by the Kardar-Parisi-Zhang equation in 1 + 1 dimensions, in the long time t limit. We obtain exact results for the related problem of p non-intersecting Brownian bridges where we compute the joint pdf Pp(M, τM ), for any finite p, where τM is the time at which the maximal height M is reached. This yields an approximation of Pt(M, XM ) for the interface problem, whose accuracy is systematically improved as p is increased, becoming exact for p → ∞. We show that our results, for moderate values of p ∼ 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 medium with one fixed endpoint.

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“…(8) over τ M , one checks that we recover the formula for the pdf of M , as obtained in Ref. [26]. Indeed, one finds Proba…”

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confidence: 64%

Extremal statistics of curved growing interfaces in 1+1 dimensions

Rambeau

Schehr

2010

EPL

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“…It seems to be highly nontrivial to extend his method to evaluate the asymptotics of higher moments h N (2n) s , s ≥ 2 for N = 2 and N ≥ 3. See the paper by Feierl on the recent progress in this combinatorial method [9]. Here we propose a different method to calculate the dominant terms of all moments of height for 2-watermelons with a wall.…”

Section: X(t) ≡ |B(t)|mentioning

confidence: 99%

“…We will perform the diffusion scaling limit first. Following the argument of [15,16,12], we can prove that the diffusion scaling limit of the N-watermelons with a wall provides the noncolliding system of N Bessel bridges, X = ( 9) where…”

Section: X(t) ≡ |B(t)|mentioning

confidence: 99%

Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function

2008

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It is known that the moments of the maximum value of a one-dimensional conditional Brownian motion, the three-dimensional Bessel bridge with duration 1 started from the origin, are expressed using the Riemann zeta function. We consider a system of two Bessel bridges, in which noncolliding condition is imposed. We show that the moments of the maximum value is then expressed using the double Dirichlet series, or using the integrals of products of the Jacobi theta functions and its derivatives. Since the present system will be provided as a diffusion scaling limit of a version of vicious walker model, the ensemble of 2-watermelons with a wall, the dominant terms in longtime asymptotics of moments of height of 2-watermelons are completely determined. For the height of 2-watermelons with a wall, the average value was recently studied by Fulmek by a method of enumerative combinatorics.

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“…The (marginal) distribution of M has recently been studied by several authors [28][29][30][31][32][33][34], while we have announced exact results for this joint distribution in the two first cases (for bridges and excursions) in a recent letter [35]. The goal of this paper is to give a detailed account of the method and the computations leading to these results.…”

Section: Introduction and Motivationsmentioning

confidence: 94%

Distribution of the time at whichNvicious walkers reach their maximal height

Rambeau

Schehr

2011

Phys. Rev. E

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We study the extreme statistics of N nonintersecting 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 nonintersecting Brownian bridges ("watermelons without wall") and nonintersecting Brownian excursions ("watermelons with a wall"). We discuss in detail the relationships between such vicious walkers models in watermelon 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)].

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The Height and Range of Watermelons without Wall

Cited by 8 publications

References 15 publications

Extremal statistics of curved growing interfaces in 1+1 dimensions

Extremal statistics of curved growing interfaces in 1+1 dimensions

Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function

Distribution of the time at whichNvicious walkers reach their maximal height

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