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Jana Reker

Publications and source records attributed to Jana Reker.

11 recordsLinked to original sources
arXiv2026

Une brève histoire des perturbations non-hermitiennes de rang un

Les perturbations de faible rang de matrices aléatoires ont été au cœur de nombreux travaux ces vingt dernières années. En particulier, les cas non-hermitiens, moins représentés dans la littérature en règle générale, font ici l'objet d'une attention spéciale en raison de leurs applications à la physique et à l'étude des réseaux de neurones. Petit tour d'horizon. -- A brief history of non-Hermitian perturbations of rank one: Low-rank perturbations of random matrices have been the focus of active research over the past twenty years. We give an overview of different non-Hermitian models, which are generally less represented in the literature, as well as some of their applications in physics and the study of neural networks.

math.PR↗
arXiv2025

LDP for the largest eigenvalue of Kronecker random matrices

We prove a large deviations principle for the largest eigenvalue of Gaussian Kronecker matrices, namely matrices defined as the sum of tensors of independent Gaussian matrices in the regime where the dimension of the Gaussian matrices goes to infinity.

math.PR↗
arXiv2023

Prethermalization for Deformed Wigner Matrices

We prove that a class of weakly perturbed Hamiltonians of the form $H_λ= H_0 + λW$, with $W$ being a Wigner matrix, exhibits prethermalization. That is, the time evolution generated by $H_λ$ relaxes to its ultimate thermal state via an intermediate prethermal state with a lifetime of order $λ^{-2}$. 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 $H_λ$.

math-ph↗
arXiv2023

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, Peché, 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].

math.PR↗
arXiv2023

Multi-Point Functional Central Limit Theorem for Wigner Matrices

Consider the random variable $\mathrm{Tr}( f_1(W)A_1\dots f_k(W)A_k)$ where $W$ is an $N\times N$ Hermitian Wigner matrix, $k\in\mathbb{N}$, and choose (possibly $N$-dependent) regular functions $f_1,\dots, f_k$ as well as bounded deterministic matrices $A_1,\dots,A_k$. 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 $f_1,\dots,f_k$ and the number of traceless matrices among $A_1,\dots,A_k$, thus extending the results of [Cipolloni, Erdős, Schröder 2023] to products of arbitrary length $k\geq2$. As an application, we consider the fluctuation of $\mathrm{Tr}(\mathrm{e}^{\mathrm{i} tW}A_1\mathrm{e}^{-\mathrm{i} tW}A_2)$ around its thermal value $\mathrm{Tr}(A_1)\mathrm{Tr}(A_2)$ when $t$ is large and give an explicit formula for the variance.

math.PR↗
arXiv2023

Dynamics of a rank-one multiplicative perturbation of a unitary matrix

We provide a dynamical study of a model of multiplicative perturbation of a unitary matrix introduced by Fyodorov. In particular, we identify a flow of deterministic domains that bound the spectrum with high probability, separating the outlier from the typical eigenvalues at all sub-critical timescales. These results are obtained under generic assumptions on $U$ that hold for a variety of unitary random matrix models.

math.PR↗
arXiv2021

Continuity properties and the support of killed exponential functionals

For two independent Lévy processes $ξ$ and $η$ and an exponentially distributed random variable $τ$ with parameter $q>0$, independent of $ξ$ and $η$, the killed exponential functional is given by $V_{q,ξ,η} := \int_0^τ\mathrm{e}^{-ξ_{s-}} \, \mathrm{d} η_s$. Interpreting the case $q=0$ as $τ=\infty$, the random variable $V_{q,ξ,η}$ is a natural generalization of the exponential functional $\int_0^\infty \mathrm{e}^{-ξ_{s-}} \, \mathrm{d} η_s$, 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 we show that also the law of the killed exponential functional $V_{q,ξ,η}$ arises as a stationary distribution of a solution to a stochastic differential equation, thus establishing a close connection to generalised Ornstein-Uhlenbeck processes. Moreover, the support and continuity of the law of killed exponential functionals is characterised, and many sufficient conditions for absolute continuity are derived. We also obtain various new sufficient conditions for absolute continuity of $\smash{\int_0^t\mathrm{e}^{-ξ_{s-}}\mathrm{d}η_s}$ for fixed $t\geq0$, as well as for integrals of the form $\smash{\int_0^\infty f(s) \, \mathrm{d}η_s}$ for deterministic functions $f$. Furthermore, applying the same techniques to the case $q=0$, new results on the absolute continuity of the improper integral $\int_0^\infty \mathrm{e}^{-ξ_{s-}} \, \mathrm{d} η_s$ are derived.

math.PR↗
arXiv2020

Short-time behavior of solutions to Lévy-driven SDEs

We consider solutions of Lévy-driven stochastic differential equations of the form $\mathrm{d} X_t=σ(X_{t-})\mathrm{d} L_t$, $X_0=x$ where the function $σ$ is twice continuously differentiable and maximal of linear growth and the driving Lévy process $L=(L_t)_{t\geq0}$ is either vector or matrix-valued. While the almost sure short-time behavior of Lévy 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 $X$. Using methods from stochastic calculus, we derive limiting results for stochastic integrals of the from $\smash{t^{-p}\int_{0+}^tσ(X_{t-})\mathrm{d} L_t}$ to show that the behavior of the quantity $t^{-p}(X_t-X_0)$ for $t\downarrow0$ almost surely mirrors the behavior of $t^{-p}L_t$. Generalizing $t^p$ to a suitable function $f:[0,\infty)\rightarrow\mathbb{R}$ then yields a tool to derive explicit LIL-type results for the solution from the behavior of the driving Lévy process.

math.PR↗
arXiv2020

On the law of killed exponential functionals

For two independent Lévy processes $ξ$ and $η$ and an exponentially distributed random variable $τ$ with parameter $q>0$ that is independent of $ξ$ and $η$, the killed exponential functional is given by $V_{q,ξ,η} := \int_0^τ\mathrm{e}^{-ξ_{s-}} \, \mathrm{d} η_s$. 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 $V_{q,ξ,η}$, 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 $q=0$ as $τ=\infty$ leads to the classical exponential functional $\int_0^\infty \mathrm{e}^{-ξ_{s-}} \, \mathrm{d} η_s$, allowing to extend many previous results to include killing.

math.PR↗
arXiv2018

Existence of Resonances for the Spin-Boson-Model with Critical Coupling Function

A two-level atom coupled to the quantized radiation field is studied. In the physical relevant situation, the coupling function modeling the interaction between the two component behaves like $|k|^{-1/2}$, as the photon momentum tends to zero. This behavior is referred to as critical, as it constitutes a borderline case. Previous results on non-existence state that, in the general case, neither a ground state nor a resonance exists. Hasler and Herbst have shown [10], however, that a ground state does exist if the absence of self-interactions is assumed. Bach, Ballesteros, Könenberg, and Menrath have then explicitly constructed the ground state this specific case [2] using the multiscale analysis known as Pizzo's Method [13]. Building on this result, the existence of resonances is considered. In the present paper, using multiscale analysis, a resonance eigenvalue of the complex deformed Hamiltonian is constructed. Neumann series expansions are used in the analysis and a suitable Feshbach-Schur map controls the exponential decay of the terms in the series.

math-ph↗