193 research outputs found
Noncolliding Squared Bessel Processes
We consider a particle system of the squared Bessel processes with index conditioned never to collide with each other, in which if
the origin is assumed to be reflecting. When the number of particles is finite,
we prove for any fixed initial configuration that this noncolliding diffusion
process is determinantal in the sense that any multitime correlation function
is given by a determinant with a continuous kernel called the correlation
kernel. When the number of particles is infinite, we give sufficient conditions
for initial configurations so that the system is well defined. There the
process with an infinite number of particles is determinantal and the
correlation kernel is expressed using an entire function represented by the
Weierstrass canonical product, whose zeros on the positive part of the real
axis are given by the particle-positions in the initial configuration. From the
class of infinite-particle initial configurations satisfying our conditions, we
report one example in detail, which is a fixed configuration such that every
point of the square of positive zero of the Bessel function is
occupied by one particle. The process starting from this initial configuration
shows a relaxation phenomenon converging to the stationary process, which is
determinantal with the extended Bessel kernel, in the long-term limit.Comment: v3: LaTeX2e, 26 pages, no figure, corrections made for publication in
J. Stat. Phy
The polyanalytic Ginibre ensembles
We consider a polyanalytic generalization of the Ginibre ensemble. This
models allowing higher Landau levels (the Ginibre ensemble corresponds to the
lowest Landau level). We study the local behavior of this point process under
blow-ups.Comment: 31 page
A real quaternion spherical ensemble of random matrices
One can identify a tripartite classification of random matrix ensembles into
geometrical universality classes corresponding to the plane, the sphere and the
anti-sphere. The plane is identified with Ginibre-type (iid) matrices and the
anti-sphere with truncations of unitary matrices. This paper focusses on an
ensemble corresponding to the sphere: matrices of the form \bY= \bA^{-1} \bB,
where \bA and \bB are independent matrices with iid standard
Gaussian real quaternion entries. By applying techniques similar to those used
for the analogous complex and real spherical ensembles, the eigenvalue jpdf and
correlation functions are calculated. This completes the exploration of
spherical matrices using the traditional Dyson indices .
We find that the eigenvalue density (after stereographic projection onto the
sphere) has a depletion of eigenvalues along a ring corresponding to the real
axis, with reflective symmetry about this ring. However, in the limit of large
matrix dimension, this eigenvalue density approaches that of the corresponding
complex ensemble, a density which is uniform on the sphere. This result is in
keeping with the spherical law (analogous to the circular law for iid
matrices), which states that for matrices having the spherical structure \bY=
\bA^{-1} \bB, where \bA and \bB are independent, iid matrices the
(stereographically projected) eigenvalue density tends to uniformity on the
sphere.Comment: 25 pages, 3 figures. Added another citation in version
Moderate deviations via cumulants
The purpose of the present paper is to establish moderate deviation
principles for a rather general class of random variables fulfilling certain
bounds of the cumulants. We apply a celebrated lemma of the theory of large
deviations probabilities due to Rudzkis, Saulis and Statulevicius. The examples
of random objects we treat include dependency graphs, subgraph-counting
statistics in Erd\H{o}s-R\'enyi random graphs and -statistics. Moreover, we
prove moderate deviation principles for certain statistics appearing in random
matrix theory, namely characteristic polynomials of random unitary matrices as
well as the number of particles in a growing box of random determinantal point
processes like the number of eigenvalues in the GUE or the number of points in
Airy, Bessel, and random point fields.Comment: 24 page
Fluctuation properties of the TASEP with periodic initial configuration
We consider the joint distributions of particle positions for the continuous
time totally asymmetric simple exclusion process (TASEP). They are expressed as
Fredholm determinants with a kernel defining a signed determinantal point
process. We then consider certain periodic initial conditions and determine the
kernel in the scaling limit. This result has been announced first in a letter
by one of us and here we provide a self-contained derivation. Connections to
last passage directed percolation and random matrices are also briefly
discussed.Comment: 33 pages, 4 figure, LaTeX; We added several references to the general
framework and techniques use
Determinantal point processes associated with Hilbert spaces of holomorphic functions
International audienceWe study determinantal point processes on C induced by the reproducing kernels of generalized Fock spaces as well as those on the unit disc D induced by the reproducing kernels of generalized Bergman spaces. In the first case, we show that all reduced Palm measures of the same order are equivalent. The Radon-Nikodym derivatives are computed explicitly using regularized multiplicative functionals. We also show that these determinantal point processes are rigid in the sense of Ghosh and Peres, hence reduced Palm measures of different orders are singular. In the second case, we show that all reduced Palm measures, of all orders, are equivalent. The Radon-Nikodym derivatives are computed using regularized multiplicative function-als associated with certain Blaschke products. The quasi-invariance of these deter-minantal point processes under the group of diffeomorphisms with compact supports follows as a corollary
Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights
We study a model of non-intersecting squared Bessel processes in the
confluent case: all paths start at time at the same positive value , remain positive, and are conditioned to end at time at . In
the limit , after appropriate rescaling, the paths fill out a
region in the -plane that we describe explicitly. In particular, the paths
initially stay away from the hard edge at , but at a certain critical
time the smallest paths hit the hard edge and from then on are stuck to
it. For we obtain the usual scaling limits from random matrix
theory, namely the sine, Airy, and Bessel kernels. A key fact is that the
positions of the paths at any time constitute a multiple orthogonal
polynomial ensemble, corresponding to a system of two modified Bessel-type
weights. As a consequence, there is a matrix valued
Riemann-Hilbert problem characterizing this model, that we analyze in the large
limit using the Deift-Zhou steepest descent method. There are some novel
ingredients in the Riemann-Hilbert analysis that are of independent interest.Comment: 59 pages, 11 figure
Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices
We study the fluctuations of eigenvalues from a class of Wigner random
matrices that generalize the Gaussian orthogonal ensemble. We begin by
considering an matrix from the Gaussian orthogonal ensemble (GOE)
or Gaussian symplectic ensemble (GSE) and let denote eigenvalue number
. Under the condition that both and tend to infinity with , we
show that is normally distributed in the limit. We also consider the
joint limit distribution of eigenvalues from the GOE or GSE with similar
conditions on the indices. The result is an -dimensional normal
distribution. Using a recent universality result by Tao and Vu, we extend our
results to a class of Wigner real symmetric matrices with non-Gaussian entries
that have an exponentially decaying distribution and whose first four moments
match the Gaussian moments.Comment: 21 pages, to appear, J. Stat. Phys. References and other corrections
suggested by the referees have been incorporate
Cell shape analysis of random tessellations based on Minkowski tensors
To which degree are shape indices of individual cells of a tessellation
characteristic for the stochastic process that generates them? Within the
context of stochastic geometry and the physics of disordered materials, this
corresponds to the question of relationships between different stochastic
models. In the context of image analysis of synthetic and biological materials,
this question is central to the problem of inferring information about
formation processes from spatial measurements of resulting random structures.
We address this question by a theory-based simulation study of shape indices
derived from Minkowski tensors for a variety of tessellation models. We focus
on the relationship between two indices: an isoperimetric ratio of the
empirical averages of cell volume and area and the cell elongation quantified
by eigenvalue ratios of interfacial Minkowski tensors. Simulation data for
these quantities, as well as for distributions thereof and for correlations of
cell shape and volume, are presented for Voronoi mosaics of the Poisson point
process, determinantal and permanental point processes, and Gibbs hard-core and
random sequential absorption processes as well as for Laguerre tessellations of
polydisperse spheres and STIT- and Poisson hyperplane tessellations. These data
are complemented by mechanically stable crystalline sphere and disordered
ellipsoid packings and area-minimising foam models. We find that shape indices
of individual cells are not sufficient to unambiguously identify the generating
process even amongst this limited set of processes. However, we identify
significant differences of the shape indices between many of these tessellation
models. Given a realization of a tessellation, these shape indices can narrow
the choice of possible generating processes, providing a powerful tool which
can be further strengthened by density-resolved volume-shape correlations.Comment: Chapter of the forthcoming book "Tensor Valuations and their
Applications in Stochastic Geometry and Imaging" in Lecture Notes in
Mathematics edited by Markus Kiderlen and Eva B. Vedel Jense
Detector Description and Performance for the First Coincidence Observations between LIGO and GEO
For 17 days in August and September 2002, the LIGO and GEO interferometer
gravitational wave detectors were operated in coincidence to produce their
first data for scientific analysis. Although the detectors were still far from
their design sensitivity levels, the data can be used to place better upper
limits on the flux of gravitational waves incident on the earth than previous
direct measurements. This paper describes the instruments and the data in some
detail, as a companion to analysis papers based on the first data.Comment: 41 pages, 9 figures 17 Sept 03: author list amended, minor editorial
change
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