30 research outputs found
Persistence probabilities in centered, stationary, Gaussian processes in discrete time
Lower bounds for persistence probabilities of stationary Gaussian processes
in discrete time are obtained under various conditions on the spectral measure
of the process. Examples are given to show that the persistence probability can
decay faster than exponentially. It is shown that if the spectral measure is
not singular, then the exponent in the persistence probability cannot grow
faster than quadratically. An example that appears (from numerical evidence) to
achieve this lower bound is presented.Comment: 9 pages; To appear in a special volume of the Indian Journal of Pure
and Applied Mathematic
Derivation of an eigenvalue probability density function relating to the Poincare disk
A result of Zyczkowski and Sommers [J.Phys.A, 33, 2045--2057 (2000)] gives
the eigenvalue probability density function for the top N x N sub-block of a
Haar distributed matrix from U(N+n). In the case n \ge N, we rederive this
result, starting from knowledge of the distribution of the sub-blocks,
introducing the Schur decomposition, and integrating over all variables except
the eigenvalues. The integration is done by identifying a recursive structure
which reduces the dimension. This approach is inspired by an analogous approach
which has been recently applied to determine the eigenvalue probability density
function for random matrices A^{-1} B, where A and B are random matrices with
entries standard complex normals. We relate the eigenvalue distribution of the
sub-blocks to a many body quantum state, and to the one-component plasma, on
the pseudosphere.Comment: 11 pages; To appear in J.Phys
Some results and problems for anisotropic random walks on the plane
This is an expository paper on the asymptotic results concerning path
behaviour of the anisotropic random walk on the two-dimensional square lattice
Z^2. In recent years Mikl\'os and the authors of the present paper investigated
the properties of this random walk concerning strong approximations, local
times and range. We give a survey of these results together with some further
problems.Comment: 20 page
Skew orthogonal polynomials and the partly symmetric real Ginibre ensemble
The partly symmetric real Ginibre ensemble consists of matrices formed as
linear combinations of real symmetric and real anti-symmetric Gaussian random
matrices. Such matrices typically have both real and complex eigenvalues. For a
fixed number of real eigenvalues, an earlier work has given the explicit form
of the joint eigenvalue probability density function. We use this to derive a
Pfaffian formula for the corresponding summed up generalized partition
function. This Pfaffian formula allows the probability that there are exactly
eigenvalues to be written as a determinant with explicit entries. It can be
used too to give the explicit form of the correlation functions, provided
certain skew orthogonal polynomials are computed. This task is accomplished in
terms of Hermite polynomials, and allows us to proceed to analyze various
scaling limits of the correlations, including that in which the matrices are
only weakly non-symmetric.Comment: 21 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
One-component plasma on a spherical annulus and a random matrix ensemble
The two-dimensional one-component plasma at the special coupling \beta = 2 is
known to be exactly solvable, for its free energy and all of its correlations,
on a variety of surfaces and with various boundary conditions. Here we study
this system confined to a spherical annulus with soft wall boundary conditions,
paying special attention to the resulting asymptotic forms from the viewpoint
of expected general properties of the two-dimensional plasma. Our study is
motivated by the realization of the Boltzmann factor for the plasma system with
\beta = 2, after stereographic projection from the sphere to the complex plane,
by a certain random matrix ensemble constructed out of complex Gaussian and
Haar distributed unitary matrices.Comment: v2, typos and references corrected, 24 pages, 1 figur
A model for the bus system in Cuernavaca (Mexico)
The bus system in Cuernavaca, Mexico and its connections to random matrix distributions have been the subject of an interesting recent study by M Krbálek and P Šeba in [15, 16]. In this paper we introduce and analyse a microscopic model for the bus system. We show that introducing a natural repulsion does produce random matrix distributions in natural double scaling regimes. The techniques employed include non-intersecting paths, logarithmic potential theory, determinantal point processes, and asymptotic analysis of several orthogonal polynomial ensembles. In addition, we introduce a circular bus model and include various calculations of non-crossing probabilities.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/48795/2/a6_28_s11.pd
Accurate measurement of thermal expansion of solids between 77 K and 350 K by 3-terminal capacitance method
Quantitative limit theorems for local functionals of arithmetic random waves
We consider Gaussian Laplace eigenfunctions on the two-dimensional flat torus
(arithmetic random waves), and provide explicit Berry-Esseen bounds in the
1-Wasserstein distance for the normal and non-normal high-energy approximation
of the associated Leray measures and total nodal lengths, respectively. Our
results provide substantial extensions (as well as alternative proofs) of
findings by Oravecz, Rudnick and Wigman (2007), Krishnapur, Kurlberg and Wigman
(2013), and Marinucci, Peccati, Rossi and Wigman (2016). Our techniques involve
Wiener-Ito chaos expansions, integration by parts, as well as some novel
estimates on residual terms arising in the chaotic decomposition of geometric
quantities that can implicitly be expressed in terms of the coarea formula.Comment: 28 page