Watermelon configurations with wall interaction: exact and asymptotic results

Abstract

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 nonrigorous) 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.2000 Mathematics Subject Classification. Primary 82B20; Secondary 05A15 05A16 60G50 82B23 82B26 82B41 82B43.

Determinants of (generalised) Catalan numbers

“…5.3] and in [28,26]; for a more detailed historical account see Footnote 6 in [32]). Since, as is the case in most applications, we do not need the theorem in its most general form, we state the special case that serves our purposes.…”

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

“…nonintersecting random walks) are fixed to the sites located near to the origin. When we impose the condition to stay positive for all vicious walkers, we say "with a wall" (at the origin) [3,8,21,12,20,22]. The height of N-watermelon is the maximum site visited by the vicious walker, who walks the farthest path from the origin.…”

Maximum distributions of bridges of noncolliding Brownian paths

“…(14) is the same as the probability density of the eigenvalue-distribution of random matrices in the class C (with variance t (1 − t)). The exponent of the factor h −N (2N +1) in (27) is the dimension of the space H C (2N). Another evidence to show the hidden symmetry of the present maximum-value problem is the following.…”

“…, n N ) ∈ N N is introduced. Though the variables n are auxiliary, since the physical quantities are given by the summations over n's as shown in (21), (27), (30) and ( (14) given in the form of that of eigenvalues of random matrices in class C and the distribution function of the maximum value (27) given in the form of the "partition function" of discrete variables implies some duality relation. The maximum-and minimum-value problems of watermelons without wall recently studied by Feierl [57] and by Schehr et al [32] are very interesting.…”

“…In particular, the version of the vicious RW model, whose continuum limit is the noncolliding Bessel bridges, r (N ) (t), 0 ≤ t ≤ 1, is called the N-watermelons with a wall [9,21,22,26,27].…”