524 research outputs found
Finite random coverings of one-complexes and the Euler characteristic
This article presents an algebraic topology perspective on the problem of
finding a complete coverage probability of a one dimensional domain by a
random covering, and develops techniques applicable to the problem beyond the
one dimensional case. In particular we obtain a general formula for the chance
that a collection of finitely many compact connected random sets placed on
has a union equal to . The result is derived under certain topological
assumptions on the shape of the covering sets (the covering ought to be {\em
good}, which holds if the diameter of the covering elements does not exceed a
certain size), but no a priori requirements on their distribution. An upper
bound for the coverage probability is also obtained as a consequence of the
concentration inequality. The techniques rely on a formulation of the coverage
criteria in terms of the Euler characteristic of the nerve complex associated
to the random covering.Comment: 25 pages,2 figures; final published versio
Effect of Dimensionality on the Percolation Thresholds of Various -Dimensional Lattices
We show analytically that the , and Pad{\'e}
approximants of the mean cluster number for site and bond percolation on
general -dimensional lattices are upper bounds on this quantity in any
Euclidean dimension , where is the occupation probability. These results
lead to certain lower bounds on the percolation threshold that become
progressively tighter as increases and asymptotically exact as becomes
large. These lower-bound estimates depend on the structure of the
-dimensional lattice and whether site or bond percolation is being
considered. We obtain explicit bounds on for both site and bond
percolation on five different lattices: -dimensional generalizations of the
simple-cubic, body-centered-cubic and face-centered-cubic Bravais lattices as
well as the -dimensional generalizations of the diamond and kagom{\'e} (or
pyrochlore) non-Bravais lattices. These analytical estimates are used to assess
available simulation results across dimensions (up through in some
cases). It is noteworthy that the tightest lower bound provides reasonable
estimates of in relatively low dimensions and becomes increasingly
accurate as grows. We also derive high-dimensional asymptotic expansions
for for the ten percolation problems and compare them to the
Bethe-lattice approximation. Finally, we remark on the radius of convergence of
the series expansion of in powers of as the dimension grows.Comment: 37 pages, 5 figure
Local Euler-Maclaurin formula for polytopes
We give a local Euler-Maclaurin formula for rational convex polytopes in a
rational euclidean space . For every affine rational polyhedral cone C in a
rational euclidean space W, we construct a differential operator of infinite
order D(C) on W with constant rational coefficients, which is unchanged when C
is translated by an integral vector. Then for every convex rational polytope P
in a rational euclidean space V and every polynomial function f (x) on V, the
sum of the values of f(x) at the integral points of P is equal to the sum, for
all faces F of P, of the integral over F of the function D(N(F)).f, where we
denote by N(F) the normal cone to P along F.Comment: Revised version (July 2006) has some changes of notation and
references adde
Symmetric measures via moments
Algebraic tools in statistics have recently been receiving special attention
and a number of interactions between algebraic geometry and computational
statistics have been rapidly developing. This paper presents another such
connection, namely, one between probabilistic models invariant under a finite
group of (non-singular) linear transformations and polynomials invariant under
the same group. Two specific aspects of the connection are discussed:
generalization of the (uniqueness part of the multivariate) problem of moments
and log-linear, or toric, modeling by expansion of invariant terms. A
distribution of minuscule subimages extracted from a large database of natural
images is analyzed to illustrate the above concepts.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6144 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Tridiagonalized GUE matrices are a matrix model for labeled mobiles
It is well-known that the number of planar maps with prescribed vertex degree
distribution and suitable labeling can be represented as the leading
coefficient of the -expansion of a joint cumulant of traces of
powers of an -by- GUE matrix. Here we undertake the calculation of this
leading coefficient in a different way. Firstly, we tridiagonalize the GUE
matrix in the manner of Trotter and Dumitriu-Edelman and then alter it by
conjugation to make the subdiagonal identically equal to . Secondly, we
apply the cluster expansion technique (specifically, the
Brydges-Kennedy-Abdesselam-Rivasseau formula) from rigorous statistical
mechanics. Thirdly, by sorting through the terms of the expansion thus
generated we arrive at an alternate interpretation for the leading coefficient
related to factorizations of the long cycle . Finally, we
reconcile the group-theoretical objects emerging from our calculation with the
labeled mobiles of Bouttier-Di Francesco-Guitter.Comment: 42 pages, LaTeX, 17 figures. The present paper completely supercedes
arXiv1203.3185 in terms of methods but addresses a different proble
Ping Pong Balayage and Convexity of Equilibrium Measures
In this presentation we prove that the equilibrium measure of a finite union of intervals on the real line or arcs on the unit circle has convex density. This is true for both, the classical logarithmic case, and the Riesz case. The electrostatic interpretation is the following: if we have a finite union of subintervals on the real line, or arcs on the unit circle, the electrostatic distribution of many “electrons” will have convex density on every subinterval. Applications to external field problems and constrained energy problems are presented
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