1,459 research outputs found
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
Magic squares and the symmetric group
Diaconis and Gamburd computed moments of secular coefficients in the CUE
ensemble. We use the characteristic map to give a new combinatorial proof of
their result. We also extend their computation to moments of traces of
symmetric powers, where the same result holds but in a wider range.Comment: 5 page
Enumerating contingency tables via random permanents
Given m positive integers R=(r_i), n positive integers C=(c_j) such that sum
r_i = sum c_j =N, and mn non-negative weights W=(w_{ij}), we consider the total
weight T=T(R, C; W) of non-negative integer matrices (contingency tables)
D=(d_{ij}) with the row sums r_i, column sums c_j, and the weight of D equal to
prod w_{ij}^{d_{ij}}. We present a randomized algorithm of a polynomial in N
complexity which computes a number T'=T'(R,C; W) such that T' < T < alpha(R, C)
T' where alpha(R,C) = min{prod r_i! r_i^{-r_i}, prod c_j! c_j^{-c_j}} N^N/N!.
In many cases, ln T' provides an asymptotically accurate estimate of ln T. The
idea of the algorithm is to express T as the expectation of the permanent of an
N x N random matrix with exponentially distributed entries and approximate the
expectation by the integral T' of an efficiently computable log-concave
function on R^{mn}. Applications to counting integer flows in graphs are also
discussed.Comment: 19 pages, bounds are sharpened, references are adde
Combinatorics and Geometry of Transportation Polytopes: An Update
A transportation polytope consists of all multidimensional arrays or tables
of non-negative real numbers that satisfy certain sum conditions on subsets of
the entries. They arise naturally in optimization and statistics, and also have
interest for discrete mathematics because permutation matrices, latin squares,
and magic squares appear naturally as lattice points of these polytopes.
In this paper we survey advances on the understanding of the combinatorics
and geometry of these polyhedra and include some recent unpublished results on
the diameter of graphs of these polytopes. In particular, this is a thirty-year
update on the status of a list of open questions last visited in the 1984 book
by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure
Secular Coefficients and the Holomorphic Multiplicative Chaos
We study the secular coefficients of random unitary matrices
drawn from the Circular -Ensemble, which are defined as the
coefficients of in the characteristic polynomial
. When we obtain a new class of limiting
distributions that arise when both and tend to infinity simultaneously.
We solve an open problem of Diaconis and Gamburd by showing that for ,
the middle coefficient tends to zero as . We show how the theory
of Gaussian multiplicative chaos (GMC) plays a prominent role in these problems
and in the explicit description of the obtained limiting distributions. We
extend the remarkable magic square formula of Diaconis and Gamburd for the
moments of secular coefficients to all and analyse the asymptotic
behaviour of the moments. We obtain estimates on the order of magnitude of the
secular coefficients for all and these estimates are sharp when
. These insights motivated us to introduce a new stochastic
object associated with the secular coefficients, which we call Holomorphic
Multiplicative Chaos (HMC). Viewing the HMC as a random distribution, we prove
a sharp result about its regularity in an appropriate Sobolev space. Our proofs
expose and exploit several novel connections with other areas, including random
permutations, Tauberian theorems and combinatorics
On a conjecture of Wilf
Let n and k be natural numbers and let S(n,k) denote the Stirling numbers of
the second kind. It is a conjecture of Wilf that the alternating sum
\sum_{j=0}^{n} (-1)^{j} S(n,j) is nonzero for all n>2. We prove this conjecture
for all n not congruent to 2 and not congruent to 2944838 modulo 3145728 and
discuss applications of this result to graph theory, multiplicative partition
functions, and the irrationality of p-adic series.Comment: 18 pages, final version, accepted for publication in the Journal of
Combinatorial Theory, Series
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