3,391 research outputs found
A discrete isodiametric result: the Erd\H{o}s-Ko-Rado theorem for multisets
There are many generalizations of the Erd\H{o}s-Ko-Rado theorem. We give new
results (and problems) concerning families of -intersecting -element
multisets of an -set and point out connections to coding theory and
classical geometry. We establish the conjecture that for such
a family can have at most members
Free fermions and the classical compact groups
There is a close connection between the ground state of non-interacting
fermions in a box with classical (absorbing, reflecting, and periodic) boundary
conditions and the eigenvalue statistics of the classical compact groups. The
associated determinantal point processes can be extended in two natural
directions: i) we consider the full family of admissible quantum boundary
conditions (i.e., self-adjoint extensions) for the Laplacian on a bounded
interval, and the corresponding projection correlation kernels; ii) we
construct the grand canonical extensions at finite temperature of the
projection kernels, interpolating from Poisson to random matrix eigenvalue
statistics. The scaling limits in the bulk and at the edges are studied in a
unified framework, and the question of universality is addressed. Whether the
finite temperature determinantal processes correspond to the eigenvalue
statistics of some matrix models is, a priori, not obvious. We complete the
picture by constructing a finite temperature extension of the Haar measure on
the classical compact groups. The eigenvalue statistics of the resulting grand
canonical matrix models (of random size) corresponds exactly to the grand
canonical measure of non-interacting free fermions with classical boundary
conditions.Comment: 35 pages, 5 figures. Final versio
Multiplicative functionals on ensembles of non-intersecting paths
The purpose of this article is to develop a theory behind the occurrence of
"path-integral" kernels in the study of extended determinantal point processes
and non-intersecting line ensembles. Our first result shows how determinants
involving such kernels arise naturally in studying ratios of partition
functions and expectations of multiplicative functionals for ensembles of
non-intersecting paths on weighted graphs. Our second result shows how Fredholm
determinants with extended kernels (as arise in the study of extended
determinantal point processes such as the Airy_2 process) are equal to Fredholm
determinants with path-integral kernels. We also show how the second result
applies to a number of examples including the stationary (GUE) Dyson Brownian
motion, the Airy_2 process, the Pearcey process, the Airy_1 and Airy_{2->1}
processes, and Markov processes on partitions related to the z-measures.Comment: 32 pages, 1 figur
Airy processes and variational problems
We review the Airy processes; their formulation and how they are conjectured
to govern the large time, large distance spatial fluctuations of one
dimensional random growth models. We also describe formulas which express the
probabilities that they lie below a given curve as Fredholm determinants of
certain boundary value operators, and the several applications of these
formulas to variational problems involving Airy processes that arise in
physical problems, as well as to their local behaviour.Comment: Minor corrections. 41 pages, 4 figures. To appear as chapter in "PASI
Proceedings: Topics in percolative and disordered systems
Gibbs Ensembles of Nonintersecting Paths
We consider a family of determinantal random point processes on the
two-dimensional lattice and prove that members of our family can be interpreted
as a kind of Gibbs ensembles of nonintersecting paths. Examples include
probability measures on lozenge and domino tilings of the plane, some of which
are non-translation-invariant.
The correlation kernels of our processes can be viewed as extensions of the
discrete sine kernel, and we show that the Gibbs property is a consequence of
simple linear relations satisfied by these kernels. The processes depend on
infinitely many parameters, which are closely related to parametrization of
totally positive Toeplitz matrices.Comment: 6 figure
Planewave density interpolation methods for 3D Helmholtz boundary integral equations
This paper introduces planewave density interpolation methods for the
regularization of weakly singular, strongly singular, hypersingular and nearly
singular integral kernels present in 3D Helmholtz surface layer potentials and
associated integral operators. Relying on Green's third identity and pointwise
interpolation of density functions in the form of planewaves, these methods
allow layer potentials and integral operators to be expressed in terms of
integrand functions that remain smooth (at least bounded) regardless the
location of the target point relative to the surface sources. Common
challenging integrals that arise in both Nystr\"om and boundary element
discretization of boundary integral equation, can then be numerically evaluated
by standard quadrature rules that are irrespective of the kernel singularity.
Closed-form and purely numerical planewave density interpolation procedures are
presented in this paper, which are used in conjunction with Chebyshev-based
Nystr\"om and Galerkin boundary element methods. A variety of numerical
examples---including problems of acoustic scattering involving multiple
touching and even intersecting obstacles, demonstrate the capabilities of the
proposed technique
- âŠ