10 research outputs found
Fluctuation Moments for Regular Functions of Wigner Matrices
We compute the deterministic approximation for mixed fluctuation moments of
products of deterministic matrices and general Sobolev functions of Wigner
matrices. Restricting to polynomials, our formulas reproduce recent results of
[Male, Mingo, Pech\'e, Speicher 2022], showing that the underlying
combinatorics of non-crossing partitions and annular non-crossing permutations
continue to stay valid beyond the setting of second-order free probability
theory. The formulas obtained further characterize the variance in the
functional central limit theorem obtained recently in the companion paper
[Reker 2023].Comment: 52 pages (including appendix), 20 figure
On the operator norm of a Hermitian random matrix with correlated entries
We consider a correlated Hermitian random matrix with a
polynomially decaying metric correlation structure. By calculating the trace of
the moments of the matrix and using the summable decay of the cumulants, we
show that its operator norm is stochastically dominated by one
Multi-Point Functional Central Limit Theorem for Wigner Matrices
Consider the random variable where
is an Hermitian Wigner matrix, , and choose
(possibly -dependent) regular functions as well as bounded
deterministic matrices . We give a functional central limit
theorem showing that the fluctuations around the expectation are Gaussian.
Moreover, we determine the limiting covariance structure and give explicit
error bounds in terms of the scaling of and the number of
traceless matrices among , thus extending the results of
[Cipolloni, Erd\H{o}s, Schr\"oder 2023] to products of arbitrary length
. As an application, we consider the fluctuation of
around its thermal value when is large
and give an explicit formula for the variance.Comment: 48 pages (including appendix
Short-time behavior of solutions to L\'evy-driven SDEs
We consider solutions of L\'evy-driven stochastic differential equations of
the form , where the
function is twice continuously differentiable and maximal of linear
growth and the driving L\'evy process is either vector or
matrix-valued. While the almost sure short-time behavior of L\'evy processes is
well-known and can be characterized in terms of the characteristic triplet,
there is no complete characterization of the behavior of the process . Using
methods from stochastic calculus, we derive limiting results for stochastic
integrals of the from
to show that the behavior of the quantity for
almost surely mirrors the behavior of . Generalizing to a
suitable function then yields a tool to
derive explicit LIL-type results for the solution from the behavior of the
driving L\'evy process
On the law of killed exponential functionals
For two independent L\'{e}vy processes and and an exponentially
distributed random variable with parameter that is independent of
and , the killed exponential functional is given by . With the killed
exponential functional arising as the stationary distribution of a Markov
process, we calculate the infinitesimal generator of the process and use it to
derive different distributional equations describing the law of
, as well as functional equations for its Lebesgue density in
the absolutely continuous case. Various special cases and examples are
considered, yielding more explicit information on the law of the killed
exponential functional and illustrating the applications of the equations
obtained. Interpreting the case as leads to the classical
exponential functional , allowing to extend many previous results to include killing
Continuity properties and the support of killed exponential functionals
For two independent L\'evy processes and and an exponentially
distributed random variable with parameter , independent of
and , the killed exponential functional is given by . Interpreting the case
as , the random variable is a natural
generalization of the exponential functional , the law of which is well-studied
in the literature as it is the stationary distribution of a generalised
Ornstein-Uhlenbeck process. In this paper, the support and continuity of the
law of killed exponential functionals is characterized, and many sufficient
conditions for absolute continuity are derived. We also obtain various new
sufficient conditions for absolute continuity of
for fixed , as well as
for integrals of the form for
deterministic functions . Furthermore, applying the same techniques to the
case , new results on the absolute continuity of the improper integral
are derived. We
also show that the law of the killed exponential functional
arises as a stationary distribution of a solution to a certain stochastic
differential equation, thus establishing a close connection to generalised
Ornstein-Uhlenbeck processes
Prethermalization for Deformed Wigner Matrices
We prove that a class of weakly perturbed Hamiltonians of the form , with being a Wigner matrix, exhibits prethermalization.
That is, the time evolution generated by relaxes to its ultimate
thermal state via an intermediate prethermal state with a lifetime of order
. Moreover, we obtain a general relaxation formula, expressing
the perturbed dynamics via the unperturbed dynamics and the ultimate thermal
state. The proof relies on a two-resolvent law for the deformed Wigner matrix
.Comment: 31 pages (including appendix), 3 figure