198,500 research outputs found
Transience of a symmetric random walk in infinite measure
We consider a random walk on a second countable locally compact topological
space endowed with an invariant Radon measure. We show that if the walk is
symmetric and if every subset which is invariant by the walk has zero or
infinite measure, then one has transience in law for almost every starting
point. We then deduce a converse to Eskin-Margulis recurrence theorem.Comment: This version gives a more complete bibliographical background that
the previous ones and contains a new section (3.3) proving the transience in
law without assumption of symmetry for walks on -covers of
finite volume space
Rectangular R-transform as the limit of rectangular spherical integrals
In this paper, we connect rectangular free probability theory and spherical
integrals. In this way, we prove the analogue, for rectangular or square
non-Hermitian matrices, of a result that Guionnet and Maida proved for
Hermitian matrices in 2005. More specifically, we study the limit, as
tend to infinity, of the logarithm (divided by ) of the expectation of
, where is the real part of an entry of , is a real number, is a certain
deterministic matrix and are independent Haar-distributed orthogonal
or unitary matrices with respective sizes , . We prove
that when the singular law of converges to a probability measure ,
for small enough, this limit actually exists and can be expressed with
the rectangular R-transform of . This gives an interpretation of this
transform, which linearizes the rectangular free convolution, as the limit of a
sequence of log-Laplace transforms.Comment: 17 page
Noncentral limit theorem for the generalized Rosenblatt process
We use techniques of Malliavin calculus to study the convergence in law of a
family of generalized Rosenblatt processes with kernels defined by
parameters taking values in a tetrahedral region of \RR^q.
We prove that, as converges to a face of , the process
converges to a compound Gaussian distribution with random variance
given by the square of a Rosenblatt process of one lower rank. The convergence
in law is shown to be stable. This work generalizes a previous result of Bai
and Taqqu, who proved the result in the case and without stability
On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables
Let be a countable infinite product of copies of the
same probability space , and let be the sequence of the
coordinate projection functions from to . Let be a
possibly nonmeasurable function from to , and let . Then we can think of as a sequence of independent
but possibly nonmeasurable random variables on . Let . By the ordinary Strong Law of Large Numbers, we almost surely
have , where
and are the lower and upper expectations. We ask if anything more precise
can be said about the limit points of in the non-trivial case where
, and obtain several negative answers. For instance, the
set of points of where converges is maximally nonmeasurable:
it has inner measure zero and outer measure one
Spectral measure of heavy tailed band and covariance random matrices
We study the asymptotic behavior of the appropriately scaled and possibly
perturbed spectral measure of large random real symmetric matrices with
heavy tailed entries. Specifically, consider the N by N symmetric matrix
whose (i,j) entry is where is an infinite array of i.i.d real variables with common
distribution in the domain of attraction of an -stable law,
, and is a deterministic function. For a random diagonal
independent of and with appropriate rescaling , we
prove that the distribution of converges in
mean towards a limiting probability measure which we characterize. As a special
case, we derive and analyze the almost sure limiting spectral density for
empirical covariance matrices with heavy tailed entries.Comment: 31 pages, minor modifications, mainly in the regularity argument for
Theorem 1.3. To appear in Communications in Mathematical Physic
Liouville Quantum Gravity and KPZ
Consider a bounded planar domain D, an instance h of the Gaussian free field
on D (with Dirichlet energy normalized by 1/(2\pi)), and a constant 0 < gamma <
2. The Liouville quantum gravity measure on D is the weak limit as epsilon
tends to 0 of the measures \epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)}dz,
where dz is Lebesgue measure on D and h_\epsilon(z) denotes the mean value of h
on the circle of radius epsilon centered at z. Given a random (or
deterministic) subset X of D one can define the scaling dimension of X using
either Lebesgue measure or this random measure. We derive a general quadratic
relation between these two dimensions, which we view as a probabilistic
formulation of the KPZ relation from conformal field theory. We also present a
boundary analog of KPZ (for subsets of the boundary of D). We discuss the
connection between discrete and continuum quantum gravity and provide a
framework for understanding Euclidean scaling exponents via quantum gravity.Comment: 56 pages. Revised version contains more details. To appear in
Inventione
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