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

Vicious walkers, friendly walkers and Young tableaux II: With a wall

We derive new results for the number of star and watermelon configurations of vicious walkers in the presence of an impenetrable wall by showing that these follow from standard results in the theory of Young tableaux, and combinatorial descriptions of symmetric functions. For the problem of $n$-friendly walkers, we derive exact asymptotics for the number of stars and watermelons both in the absence of a wall and in the presence of a wall.Comment: 35 pages, AmS-LaTeX; Definitions of n-friendly walkers clarified; the statement of Theorem 4 and its proof were correcte

Extreme value distributions of noncolliding diffusion processes

Noncolliding diffusion processes reported in the present paper are $N$-particle systems of diffusion processes in one-dimension, which are conditioned so that all particles start from the origin and never collide with each other in a finite time interval $(0, T)$, $0 < T < \infty$. We consider four temporally inhomogeneous processes with duration $T$, the noncolliding Brownian bridge, the noncolliding Brownian motion, the noncolliding three-dimensional Bessel bridge, and the noncolliding Brownian meander. Their particle distributions at each time $t \in [0, T]$ are related to the eigenvalue distributions of random matrices in Gaussian ensembles and in some two-matrix models. Extreme values of paths in $[0, T]$ are studied for these noncolliding diffusion processes and determinantal and pfaffian representations are given for the distribution functions. The entries of the determinants and pfaffians are expressed using special functions.Comment: v2: LaTeX2e, 21 pages, 2 figures, correction mad

Watermelon configurations with wall interaction: exact and asymptotic results

We perform an exact and asymptotic analysis of the model of $n$ vicious walkers interacting with a wall via contact potentials, a model introduced by Brak, Essam and Owczarek. More specifically, we study the partition function of watermelon configurations which start on the wall, but may end at arbitrary height, and their mean number of contacts with the wall. We improve and extend the earlier (partially non-rigorous) results by Brak, Essam and Owczarek, providing new exact results, and more precise and more general asymptotic results, in particular full asymptotic expansions for the partition function and the mean number of contacts. Furthermore, we relate this circle of problems to earlier results in the combinatorial and statistical literature.Comment: AmS-TeX, 41 page

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

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.Comment: 37 pages, 4 figures, revised and published version. A typo has been corrected in Eq. (10

Composition schemes: q-enumerations and phase transitions

Composition schemes are ubiquitous in combinatorics, statistical mechanics and probability theory. We give a unifying explanation to various phenomena observed in the combinatorial and statistical physics literature in the context of $q$-enumeration (this is a model where objects with a parameter of value $k$ have a Gibbs measure/Boltzmann weight $q^k$). This leads to phase transition diagrams, highlighting the effect of $q$-enumeration and $q$-distributions on the nature of the limit laws. We build upon the classical distinction between subcritical, critical and supercritical composition schemes, as well as recent results. We apply our results to a wealth of parameters, adding new limit laws to various families of lattice paths and quarter plane walks. We also explain previously observed limit laws for pattern restricted permutations, and a phenomenon observed by Krattenthaler for the wall contacts in watermelons with a wall.Comment: 12 page

Random lattice walks in a Weyl chamber of type A or B and non-intersecting lattice paths

Die vorliegende Arbeit beschäftigt sich mit zwei eng verwandten Modellen: Gitterpfaden in einer Weylkammer vom Typ B und nichtüberschneidenden Gitterpfaden im ganzzahligen Gitter aufgespannt durch die Vektoren {(1,1),(1,-1)} mit Schritten aus dieser Menge. Diese Gitterpfadmodelle sind von zentraler Bedeutung z.B. in der Kombinatorik und der statistischen Mechanik. In der statistischen Mechanik dienen diese Modelle der Beschreibung bestimmter nicht-kollidierender Teilchen-Systeme. Die Bedeutung von Gitterpfadmodellen in der Kombinatorik ist teilweise begründet durch ihre interessanten kombinatorischen Eigenschaften, vor allem aber auch durch die engen Beziehungen zu zahlreichen zentralen kombinatorischen Objekten wie z.B. Integer Partitions, Plane Partitions und Young Tableaux. Im ersten Teil dieser Arbeit werden asymptotische Formeln für die Anzahl von Gitterpfaden in einer Weylkammer vom Typ B für eine allgemeine Klasse von Schritten hergeleitet. Die Klasse der zulässigen Schritte wird hierbei durch die Forderung der "Reflektierbarkeit" der resultierenden Pfade beschränkt. Spezialfälle dieser asymptotischen Formel lösen in der Literatur aufgeworfene Probleme und liefern bekannte Resultate für zweidimensionale Vicious Walkers Modelle und sogenannte k-non-crossing tangled diagrams. Im zweiten Teil werden die Zufallsvariablen "Höhe" und "Ausdehnung" auf der Menge aller nichtüberschneidenden Gitterpfade mit n Schritten sowie auf der Teilmenge all jener auf die obere Halbebene beschränkten nichtüberschneidenden Gitterpfade mit n Schritten studiert. Unter der Annahme einer Gleichverteilung auf diesen Mengen wird die asymptotische Verteilung beider Zufallsvariablen bestimmt. Weiters werden die ersten beiden Terme der asymptotischen Entwicklung aller Momente der Zufallsvariable "Höhe" ermittelt. Dies löst ein in der Literatur aufgeworfenes Problem, und verallgemeinert ein bekanntes Resultat über die Höhe ebener Wurzelbäume.This thesis is concerned with two closely related lattice walk models: lattice walks in a Weyl chamber type B and non-intersecting lattice paths on the integer lattice spanned by the vectors {(1,1),(1,-1)} with steps from this set. These models play an important role in, e.g., combinatorics and statistical mechanics. In statistical mechanics, non-intersecting lattice paths serve as models for certain non-colliding particle systems. From a combinatorial point of view, lattice paths models are very natural objects to study, partly because of their intrinsic interesting combinatorics, and partly because of their close relationship to many other important combinatorial structures, such as integer partitions, plane partitions and Young tableaux. In the first part of this thesis, we determine asymptotics for the number of lattice walks in a Weyl chamber of type B for a general class of steps. The class of admissible steps is determined by requiring the walks to be "reflectable". As special cases, these asymptotics include several results found in the literature, e.g., asymptotics for certain vicious walkers models and k-non-crossing tangled diagrams. In the second part of this thesis we study the random variables "height" and "range" on the set of non-intersecting lattice paths of length n as well as on the subset of those non-intersecting lattice paths of length n that are confined to the upper half plane. Assuming the uniform probability distribution on these sets, we determine the asymptotic distribution of both random variables as the number of steps tends to infinity as well as first and second order asymptotics for all moments of the random variable "height". This solves a problem raised in the literature, and generalises a well-known result on the height of random planted plane trees