@unpublished{64816, abstract = {{We study a block mean-field Ising model with $N$ spins split into $s_N$ blocks, with Curie-Weiss interaction within blocks and nearest-neighbor coupling between blocks. While previous models deal with the block magnetization for a fixed number of blocks, we study the the simultaneous limit $N\to\infty$ and $s_N\to\infty$. The model interpolates between Curie-Weiss model for $s_N=1$, multi-species mean field for fixed $s_N=s$, and the 1D Ising model for each spin in its own block at $s_N=N$. Under mild growth conditions on $s_N$, we prove a law of large numbers and a multivariate CLT with covariance given by the lattice Green's function. For instance, the high temperature CLT essentially covers the optimal range up to $s_N=o(N/(\log N)^c)$ and the low temperature regime is new even for fixed number of blocks $s > 2$. In addition to the standard competition between entropy and energy, a new obstacle in the proofs is a curse of dimensionality as $s_N \to \infty$.}}, author = {{Jalowy, Jonas and Lammers, Isabel and Löwe, Matthias}}, booktitle = {{arXiv:2603.01994}}, title = {{{The infinite block spin Ising model}}}, year = {{2026}}, } @unpublished{59664, abstract = {{Given a sequence of polynomials $(P_n)_{n \in \mathbb{N}}$ with only nonpositive zeros, the aim of this article is to present a user-friendly approach for determining the limiting zero distribution of $P_n$ as $\mathrm{deg}\, P_n \to \infty$. The method is based on establishing an equivalence between the existence of a limiting empirical zero distribution $\mu$ and the existence of an exponential profile $g$ associated with the coefficients of the polynomials $(P_n)_{n \in \mathbb{N}}$. The exponential profile $g$, which can be roughly described by $[z^k]P_n(z) \approx \exp(n g(k/n))$, offers a direct route to computing the Cauchy transform $G$ of $\mu$: the functions $t \mapsto tG(t)$ and $\alpha \mapsto \exp(-g'(\alpha))$ are mutual inverses. This relationship, in various forms, has previously appeared in the literature, most notably in the paper [Van Assche, Fano and Ortolani, SIAM J. Math. Anal., 1987]. As a first contribution, we present a self-contained probabilistic proof of this equivalence by representing the polynomials as generating functions of sums of independent Bernoulli random variables. This probabilistic framework naturally lends itself to tools from large deviation theory, such as the exponential change of measure. The resulting theorems generalize and unify a range of previously known results, which were traditionally established through analytic or combinatorial methods. Secondly, using the profile-based approach, we investigate how the exponential profile and the limiting zero distribution behave under certain operations on polynomials, including finite free convolutions, Hadamard products, and repeated differentiation. In particular, our approach yields new proofs of the convergence results `$\boxplus_n \to \boxplus$' and `$\boxtimes_n \to \boxtimes$', extending them to cases where the distributions are not necessarily compactly supported.}}, author = {{Jalowy, Jonas and Kabluchko, Zakhar and Marynych, Alexander}}, booktitle = {{arXiv:2504.11593}}, title = {{{Zeros and exponential profiles of polynomials I: Limit distributions, finite free convolutions and repeated differentiation}}}, year = {{2025}}, } @article{59665, author = {{Erbar, Matthias and Huesmann, Martin and Jalowy, Jonas and Müller, Bastian}}, issn = {{0022-1236}}, journal = {{Journal of Functional Analysis}}, number = {{4}}, publisher = {{Elsevier BV}}, title = {{{Optimal transport of stationary point processes: Metric structure, gradient flow and convexity of the specific entropy}}}, doi = {{10.1016/j.jfa.2025.110974}}, volume = {{289}}, year = {{2025}}, } @unpublished{60293, abstract = {{In this work, we present a complete characterization of the covariance structure of number statistics in boxes for hyperuniform point processes. Under a standard integrability assumption, the covariance depends solely on the overlap of the faces of the box. Beyond this assumption, a novel interpolating covariance structure emerges. This enables us to identify a limiting Gaussian 'coarse-grained' process, counting the number of points in large boxes as a function of the box position. Depending on the integrability assumption, this process may be continuous or discontinuous, e.g. in d=1 it is given by an increment process of a fractional Brownian motion.}}, author = {{Jalowy, Jonas and Stange, Hanna}}, booktitle = {{arXiv:2506.13661}}, title = {{{Box-Covariances of Hyperuniform Point Processes}}}, year = {{2025}}, } @article{59507, abstract = {{Differential equations posed on quadratic matrix Lie groups arise in the context of classical mechanics and quantum dynamical systems. Lie group numerical integrators preserve the constants of motions defining the Lie group. Thus, they respect important physical laws of the dynamical system, such as unitarity and energy conservation in the context of quantum dynamical systems, for instance. In this article we develop a high-order commutator free Lie group integrator for non-autonomous differential equations evolving on quadratic Lie groups. Instead of matrix exponentials, which are expensive to evaluate and need to be approximated by appropriate rational functions in order to preserve the Lie group structure, the proposed method is obtained as a composition of Cayley transforms which naturally respect the structure of quadratic Lie groups while being computationally efficient to evaluate. Unlike Cayley-Magnus methods the method is also free from nested matrix commutators.}}, author = {{Wembe Moafo, Boris Edgar and Offen, Cristian and Maslovskaya, Sofya and Ober-Blöbaum, Sina and Singh, Pranav}}, journal = {{J. Comput. Appl. Math}}, number = {{15}}, title = {{{Commutator-free Cayley methods}}}, doi = {{10.1016/j.cam.2025.117184}}, volume = {{477}}, year = {{2025}}, } @unpublished{63394, abstract = {{We study the statistics of the number of real eigenvalues in the elliptic deformation of the real Ginibre ensemble. As the matrix dimension grows, the law of large numbers and the central limit theorem for the number of real eigenvalues are well understood, but the probabilities of rare events remain largely unexplored. Large deviation type results have been obtained only in extreme cases, when either a vanishingly small proportion of eigenvalues are real or almost all eigenvalues are real. Here, in both the strong and weak asymmetry regimes, we derive the probabilities of rare events in the moderate-to-large deviation regime, thereby providing a natural connection between the previously known regime of Gaussian fluctuations and the large deviation regime. Our results are new even for the classical real Ginibre ensemble.}}, author = {{Byun, Sung-Soo and Jalowy, Jonas and Lee, Yong-Woo and Schehr, Grégory}}, booktitle = {{arXiv:2511.09191}}, title = {{{Moderate-to-large deviation asymptotics for real eigenvalues of the elliptic Ginibre matrices}}}, year = {{2025}}, } @unpublished{63393, abstract = {{We study the evolution of zeros of high polynomial powers under the heat flow. For any fixed polynomial $P(z)$, we prove that the empirical zero distribution of its heat-evolved $n$-th power converges to a distribution on the complex plane as $n$ tends to infinity. We describe this limit distribution $μ_t$ as a function of the time parameter $t$ of the heat evolution: For small time, zeros start to spread out in approximately semicircular distributions, then intricate curves start to form and merge, until for large time, the zero distribution approaches a widespread semicircle law through the initial center of mass. The Stieltjes transform of the limit distribution $μ_t$ satisfies a self-consistent equation and a Burgers' equation. The present paper deals with general complex-rooted polynomials for which, in contrast to the real-rooted case, no free-probabilistic representation for $μ_t$ is available.}}, author = {{Höfert, Antonia and Jalowy, Jonas and Kabluchko, Zakhar}}, booktitle = {{arXiv:2512.17808}}, title = {{{Zeros of polynomial powers under the heat flow}}}, year = {{2025}}, } @article{62291, abstract = {{In [Jalowy, Kabluchko, Marynych, arXiv:2504.11593v1, 2025], the authors discuss a user-friendly approach to determine the limiting empirical zero distribution of a sequence of real-rooted polynomials, as the degree goes to $\infty$. In this note, we aim to apply it to a vast range of examples of polynomials providing a unifying source for limiting empirical zero distributions. We cover Touchard, Fubini, Eulerian, Narayana and little $q$-Laguerre polynomials as well as hypergeometric polynomials including the classical Hermite, Laguerre and Jacobi polynomials. We construct polynomials whose empirical zero distributions converge to the free multiplicative normal and Poisson distributions. Furthermore, we study polynomials generated by some differential operators. As one inverse result, we derive coefficient asymptotics of the characteristic polynomial of random covariance matrices.}}, title = {{{Zeros and exponential profiles of polynomials II: Examples}}}, doi = {{10.48550/ARXIV.2509.11248}}, year = {{2025}}, } @article{53146, author = {{Berger, Thomas and Dennstädt, Dario and Lanza, L. and Worthmann, K. }}, journal = {{SIAM Journal on Control and Optimization}}, title = {{{Robust Funnel Model Predictive Control for Output Tracking with Prescribed Performance}}}, year = {{2024}}, } @article{53142, author = {{Berger, Thomas and Lanza, Lukas}}, journal = {{IMA Journal of Mathematical Control and Information,}}, number = {{4}}, pages = {{691--713}}, title = {{{Funnel control of linear systems with arbitrary relative degree under output measurement losses}}}, doi = {{doi: 10.1093/imamci/dnad029}}, volume = {{40}}, year = {{2023}}, } @article{53143, author = {{Lee, J. G. and Berger, Thomas and Trenn, S. and Shim, H.}}, journal = {{Automatica}}, pages = {{Article 111204}}, title = {{{Edge-wise funnel output synchronization of heterogeneous agents with relative degree one}}}, doi = {{doi: 10.1016/j.automatica.2023.111204 (open access)}}, volume = {{156}}, year = {{2023}}, } @article{35644, author = {{Kolb, Martin and Klump, Alexander}}, journal = {{Theory of Probability and its Applications}}, number = {{4}}, pages = {{717--744}}, publisher = {{Society for Industrial and Applied Mathematics}}, title = {{{Uniqueness of the Inverse First Passage Time Problem and the Shape of the Shiryaev boundary}}}, volume = {{67}}, year = {{2022}}, } @article{35649, abstract = {{Motivated by the work [6] of Mariusz Bieniek, Krzysztof Burdzy and Soumik Pal we study a Fleming-Viot-type particle system consisting of independently moving particles each driven by generalized Bessel processes on the positive real line. Upon hitting the boundary {0} this particle is killed and an uniformly chosen different one branches into two particles. Using the symmetry of the model and the self similarity property of Bessel processes, we obtain a criterion to decide whether the particles converge to the origin at a finite time. This addresses open problem 1.4 in [6]. Specifically, inspired by [6, Open Problem 1.5], we investigate the case of three moving particles and refine the general result of [6, Theorem 1.1(ii)] extending the regime of drift parameters, where convergence does not occur – even to values, where it does occur when considering the case of only two particles.}}, author = {{Kolb, Martin and Liesenfeld, Matthias}}, journal = {{Electronic Journal of Probability}}, number = {{27}}, pages = {{1--28}}, publisher = {{Institute of Mathematical Statistics}}, title = {{{On non-extinction in a Fleming-Viot-type particle model with Bessel drift}}}, doi = {{https://doi.org/10.1214/22-EJP866}}, year = {{2022}}, } @article{35650, abstract = {{We consider autoregressive sequences Xn = aXn−1 + ξn and Mn = max{aMn−1 , ξn} with a constant a ∈ (0, 1) and with positive, in- dependent and identically distributed innovations {ξk }. It is known that if P(ξ1 > x) ∼ d log x with some d ∈ (0, − log a) then the chains {Xn} and {Mn} are null recurrent. We investigate the tail behaviour of recurrence times in this case of logarithmically decaying tails. More precisely, we show that the tails of recurrence times are regularly varying of index −1 − d/ log a. We also prove limit theorems for {Xn} and {Mn} conditioned to stay over a fixed level x0. Furthermore, we study tail asymptotics for recurrence times of {Xn} and {Mn} in the case when these chains are positive recurrent and the tail of log ξ1 is subexponential.}}, author = {{Denisov, Denis and Hinrichs, Günter and Kolb, Martin and Wachtel, Vitali}}, journal = {{Electronic Journal of Probability}}, pages = {{1--43}}, publisher = {{Institute of Mathematical Statistics}}, title = {{{Persistence of autoregressive sequences with logarithmic tails}}}, doi = {{https://doi.org/10.48550/arXiv.2203.14772}}, volume = {{27}}, year = {{2022}}, } @article{33278, abstract = {{The kinetic Brownian motion on the sphere bundle of a Riemannian manifold M is a stochastic process that models a random perturbation of the geodesic flow. If M is an orientable compact constantly curved surface, we show that in the limit of infinitely large perturbation the L2-spectrum of the infinitesimal generator of a time-rescaled version of the process converges to the Laplace spectrum of the base manifold.}}, author = {{Kolb, Martin and Weich, Tobias and Wolf, Lasse}}, journal = {{Annales Henri Poincaré }}, number = {{4}}, pages = {{1283--1296}}, publisher = {{Springer Science + Business Media}}, title = {{{Spectral Asymptotics for Kinetic Brownian Motion on Surfaces of Constant Curvature}}}, volume = {{23}}, year = {{2021}}, } @article{33481, abstract = {{While 2D Gibbsian particle systems might exhibit orientational order resulting in a lattice-like structure, these particle systems do not exhibit positional order if the interaction between particles satisfies some weak assumptions. Here we investigate to which extent particles within a box of size may fluctuate from their ideal lattice position. We show that particles near the center of the box typically show a displacement at least of order . Thus we extend recent results on the hard disk model to particle systems with fairly arbitrary particle spins and interaction. Our result applies to models such as rather general continuum Potts type models, e.g. with Widom–Rowlinson or Lenard-Jones-type interaction.}}, author = {{Richthammer, Thomas and Fiedler, Michael}}, journal = {{Stochastic Processes and their Applications}}, pages = {{1--32}}, publisher = {{Elsevier}}, title = {{{A lower bound on the displacement of particles in 2D Gibbsian particle systems}}}, doi = {{https://doi.org/10.1016/j.spa.2020.10.003}}, volume = {{132}}, year = {{2021}}, } @misc{33273, abstract = {{Dieses Lernangebot widmet sich der linearen Algebra als dem Teil der Mathematik, der neben der Optimierung und der Stochastik die Grundlage für praktisch alle Entwicklungen im Bereich Künstliche Intelligenz (KI) darstellt. Das Fach ist jedoch für Anfänger meist ungewohnt abstrakt und wird daher oft als besonders schwierig und unanschaulich empfunden. In diesem Kurs wird das Erlernen mathematischer Kenntnisse in linearer Algebra verknüpft mit dem aktuellen und faszinierenden Anwendungsfeld der künstlichen neuronalen Netze (KNN). Daraus ergeben sich in natürlicher Weise Anwendungsbeispiele, an denen die wesentlichen Konzepte der linearen Algebra erklärt werden können. Behandelte Themen sind: Der Vektorraum der reellen Zahlentupel, reelle Vektorräume allgemein Lineare Abbildungen Matrizen Koordinaten und darstellende Matrizen Lineare Gleichungssysteme, Gaußalgorithmus Determinante Ein Ausblick auf nichtlineare Techniken, die für neuronale Netzwerke relevant sind.}}, author = {{Schramm, Thomas and Gasser, Ingenuin and Schwenker, Sören and Seiler, Ruedi and Lohse, Alexander and Zobel, Kay}}, publisher = {{Hamburg Open Online University}}, title = {{{Linear Algebra driven by Data Science}}}, year = {{2020}}, } @article{33282, abstract = {{We derive a criterium for the almost sure finiteness of perpetual integrals of L ́evy processes for a class of real functions including all continuous functions and for general one- dimensional L ́evy processes that drifts to plus infinity. This generalizes previous work of D ̈oring and Kyprianou, who considered L ́evy processes having a local time, leaving the general case as an open problem. It turns out, that the criterium in the general situation simplifies significantly in the situation, where the process has a local time, but we also demonstrate that in general our cri- terium can not be reduced. This answers an open problem posed in D ̈oring, L. and Kyprianou, A. (2015).}}, author = {{Kolb, Martin and Savov, Mladen}}, journal = {{Bernoulli}}, keywords = {{L ́evy processes, Perpetual integrals, Potential measures}}, number = {{2}}, pages = {{1453--1472}}, publisher = {{Bernoulli Society for Mathematical Statistics and Probability}}, title = {{{A Characterization of the Finiteness of Perpetual Integrals of Levy Processes}}}, doi = {{https://doi.org/10.48550/arXiv.1903.03792}}, volume = {{26}}, year = {{2020}}, } @article{33330, abstract = {{Reciprocal relations are binary relations Q with entries Q(i,j)∈[0,1], and such that Q(i,j)+Q(j,i)=1. Relations of this kind occur quite naturally in various domains, such as preference modeling and preference learning. For example, Q(i,j) could be the fraction of voters in a population who prefer candidate i to candidate j. In the literature, various attempts have been made at generalizing the notion of transitivity to reciprocal relations. In this paper, we compare three important frameworks of generalized transitivity: g-stochastic transitivity, T-transitivity, and cycle-transitivity. To this end, we introduce E-transitivity as an even more general notion. We also use this framework to extend an existing hierarchy of different types of transitivity. As an illustration, we study transitivity properties of probabilities of pairwise preferences, which are induced as marginals of an underlying probability distribution on rankings (strict total orders) of a set of alternatives. In particular, we analyze the interesting case of the so-called Babington Smith model, a parametric family of distributions of that kind.}}, author = {{Haddenhorst, Björn and Hüllermeier, Eyke and Kolb, Martin}}, journal = {{International Journal of Approximate Reasoning}}, number = {{2}}, pages = {{373--407}}, publisher = {{Elsevier}}, title = {{{Generalized transitivity: A systematic comparison of concepts with an application to preferences in the Babington Smith model}}}, doi = {{https://doi.org/10.1016/j.ijar.2020.01.007}}, volume = {{119}}, year = {{2020}}, } @article{33331, abstract = {{Motivated by the recent contribution (Bauer and Bernard in Annales Henri Poincaré 19:653–693, 2018), we study the scaling limit behavior of a class of one-dimensional stochastic differential equations which has a unique attracting point subject to a small additional repulsive perturbation. Problems of this type appear in the analysis of continuously monitored quantum systems. We extend the results of Bauer and Bernard (Annales Henri Poincaré 19:653–693, 2018) and prove a general result concerning the convergence to a homogeneous Poisson process using only classical probabilistic tools.}}, author = {{Kolb, Martin and Liesenfeld, Matthias}}, journal = {{Annales Henri Poincaré}}, number = {{6}}, pages = {{1753--1783}}, publisher = {{Institute Henri Poincaré}}, title = {{{Stochastic Spikes and Poisson Approximation of One-Dimensional Stochastic Differential Equations with Applications to Continuously Measured Quantum Systems}}}, doi = {{http://dx.doi.org/10.1007/s00023-019-00772-9}}, volume = {{20}}, year = {{2019}}, }