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 long-time 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.
Deep Dive into Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function.
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 long-time 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.
Let B(t) = (B 1 (t), B 2 (t), B 3 (t)), t ≥ 0 be the three-dimensional Brownian motion (BM), in which three components B j (t), j = 1, 2, 3 are given by independent one-dimensional standard BMs. The three-dimensional Bessel process (BES 3 ), X(t), started from x > 0 is defined as the radial part of B(t), with X(0) = x. BES 3 is a diffusion process on R + = {x ∈ R : x ≥ 0}, where R denotes the set of all real numbers. By Itô's formula we can show that it satisfies the stochastic differential equation of the form dX(t) = dB(t) + 1 X(t) dt, t ≥ 0, X(0) = x, where B(t) is the one-dimensional standard BM different from B j (t)'s used to give B(t) above. We can prove that X(t) → ∞ in t → ∞ with probability one for all x ≥ 0, i.e. BES 3 is transient. For the basic properties of BES 3 , see, for example, 3.3 C in [13], VI.3 in [23], IV.34 in [6].
The three-dimensional Bessel bridge with duration 1 started from the origin, X(t), t ∈ [0, 1], is then defined as the BES 3 conditioned x = X(0) = 0 and X(1) = 0.
Figure 1 illustrates a sample path of X(t) on the spatio-temporal plane (t, x) ∈ [0, 1]×R + . In [4], a variety of probability laws associated with conditional Brownian motions are discussed, which are related to the Jacobi theta function and the Riemann zeta function. One of them is the probability law of the maximum value of X(t);
(1.1)
Let E[H s 1 ] be the s-th moment of H 1 . The following equality is discussed in [4],
where C denotes the set of all complex numbers, and
with the gamma function
and with the Riemann zeta function
See also Chapter 11 in [25].
We know the two facts; (i) the BM can be realized as the diffusion scaling limit of the simple random walk, (ii) the probability law of BES 3 is equal to that of the BM conditioned to stay positive. Combination of them will lead to the following. For a fixed n ≥ 1, consider one-dimensional simple random walks started from the origin, which visit only positive sites {1, 2, 3, • • •} up to time 2n and return to the origin at time 2n. Sample paths of such conditional random walks are called Dyck paths of length n in combinatorics. The height of Dyck path h 1 (2n) is defined as the maximum site visited by the walker. Let • denote the average over all Dyck paths with uniform weight. Then we will have the relation
The classical work of de Bruijn, Knuth and Rice in enumerative combinatorics [7] gives
Here we should note that, through the relations (1.2) and (1.4), if we only consider the dominant term in (1.5) proportional to √ n, this result in combinatorics means nothing but the fact ξ(1) = 1/2. It is rather obvious if we know the following integral representation of ξ(s) due to Riemann,
where ϑ(u) is a version of the Jacobi theta function
Recently Fulmek reported a generalization of the result of de Bruijn, Knuth and Rice, by calculating the asymptotics of the average height of 2-watermelons with a wall [11]. In general, the uniform ensemble of N-watermelons, N ≥ 2, is a version of vicious walker model of Fisher [10]. In this version the starting points and the ending points of N vicious walkers (i.e. 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. Let h 2 (2n) be the height of 2-watermelon with a wall. Fulmek showed
Here the factor c 2 = 2.57758 • • • of the dominant term proportional to √ n was given by numerical evaluation of the “constant terms” in Laurent expansions of a version of double Dirichlet series. The terms are represented by integrals of functions expressed using the Jacobi theta function (1.7) and its derivatives. It should be emphasized the fact that Fulmek succeeded in proving the N = 2 case of the conjecture of Bonichon and Mosbah [5],
obtained by computer simulations for the average height h N (2n) of general N-watermelons with a wall, N ≥ 1. 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. 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,
where
In the present paper we determine E[H s 2 ] for arbitrary s for the two Bessel bridges with noncolliding condition. Noncolliding diffusion particle systems are interesting and important statistical-mechanical processes, since they are related to the group representation-theory, the random matrix theory, and the exactly solved nonequilibrium statis
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
