3,236 research outputs found
Analyzing Boltzmann Samplers for Bose-Einstein Condensates with Dirichlet Generating Functions
Boltzmann sampling is commonly used to uniformly sample objects of a
particular size from large combinatorial sets. For this technique to be
effective, one needs to prove that (1) the sampling procedure is efficient and
(2) objects of the desired size are generated with sufficiently high
probability. We use this approach to give a provably efficient sampling
algorithm for a class of weighted integer partitions related to Bose-Einstein
condensation from statistical physics. Our sampling algorithm is a
probabilistic interpretation of the ordinary generating function for these
objects, derived from the symbolic method of analytic combinatorics. Using the
Khintchine-Meinardus probabilistic method to bound the rejection rate of our
Boltzmann sampler through singularity analysis of Dirichlet generating
functions, we offer an alternative approach to analyze Boltzmann samplers for
objects with multiplicative structure.Comment: 20 pages, 1 figur
Special values of multiple polylogarithms
Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier
Resolution of Some Open Problems Concerning Multiple Zeta Evaluations of Arbitrary Depth
We prove some new evaluations for multiple polylogarithms of arbitrary depth.
The simplest of our results is a multiple zeta evaluation one order of
complexity beyond the well-known Broadhurst-Zagier formula. Other results we
provide settle three of the remaining outstanding conjectures of Borwein,
Bradley, and Broadhurst. A complete treatment of a certain arbitrary depth
class of periodic alternating unit Euler sums is also given.Comment: 21 pages, To appear in Compositio Mathematic
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.Comment: v2: LaTeX, 19 pages, 2 figures, minor corrections made for
publication in J. Stat. Phy
Pseudomoments of the Riemann zeta-function and pseudomagic squares
We compute integral moments of partial sums of the Riemann zeta function on
the critical line and obtain an expression for the leading coefficient as a
product of the standard arithmetic factor and a geometric factor. The geometric
factor is equal to the volume of the convex polytope of substochastic matrices
and is equal to the leading coefficient in the expression for moments of
truncated characteristic polynomial of a random unitary matrix
- …