GRK Seminar

In this talk, we discuss some limitations of the quantum graph approach in defining Dirichlet forms, Laplacians and/or Brownian motions in case when the graph has vanishing edge lengths. Our model set is the stretched Sierpinski gasket, SSG for short. It is the space obtained by replacing every branching point of the Sierpinski gasket by an interval. (It has also been called "Hanoi attractor".) As a result, it is the closure of a countable union of intervals and one might expect that a diffusion on SSG is essentially a kind of gluing together the Brownian motions on the intervals. In fact, there have been several works in this direction. However, there still remains, "reminiscence" of the Sierpinski gasket in the geometric as well as analytic structure of SSG and the same should therefore be expected for diffusions. This is a joint work with Patricia Alonso Ruiz (Texas A&M University) and Jun Kigami (Kyoto University).

The assumption that high dimensional data is Gaussian is pervasive in many statistical procedures, due to the ease of analytic tractability this special distribution provides. We explore the relaxation of the Gaussian assumption in Shrinkage estimation, Stein's Unbiased Risk Estimate, and Single Index Models, using two tools that originate in Stein's method: Stein kernels, and the zero bias distribution. Taking this approach leads to measures of discrepancy from the Gaussian that arise naturally from the form of the procedures considered, and result in performance bounds that apply in contexts not restricted to the Gaussian. The resulting bounds typically include an additional term that reflects the cost of deviation from the Gaussian, and that vanishes for the Gaussian, thus recovering this paricular special case. (with Xiaohan Wei, Max Fathi, Gesine Reinert, and Adrien Samaurd)

A great achievement of physics in the second half of the twentieth century has been the prediction of conformal symmetry of the scaling limit of critical statistical physics systems. Around the turn of the millennium, the mathematical understanding of this fact has progressed tremendously in two dimensions with the introduction of the Schramm-Loewner Evolution and the proofs of conformal invariance of the Ising model and dimers. Nevertheless, the understanding is still restricted to very specific models. In this talk, we will gently introduce the notion of conformal invariance of lattice systems by taking the example of percolation models. We will also explain some recent and partial progress in the direction of proving conformal invariance for a large class of such models.

Consider Bessel, Jacobi, and Dunkl processes on $\mathbb{R}^N$ which depend on several positive coupling constants $k$. These processes describe interacting particle systems of Calogero-Moser-Sutherland type with $N$ particles. For some cases, these processes are related with $\beta$-Hermite, -Laguerre, and -Jacobi ensembles. Moreover, for the frozen case $k=\infty$, the processes degenerate to deterministic or pure jump processes. We show how the generators for these processes lead to analogues of Wigner's semicircle and Marchenko-Pastur limit laws for $N\to\infty$ for the empirical distributions of the particles with arbitrary initial empirical distributions by using free convolutions. In the frozen cases, our approach leads to a new proof of well known limit laws for the empirical measures of the zeroes of Hermite, Laguerre, and Jacobi polynomials. The talk is based on joint work with J. Woerner and M. Auer.

Disordered systems such as glasses and spin glasses pose a challenge to theoretical physics. As a simplification mean-field models where the geometry of interactions is induced by a random graph have been introduced. These models turn out to be intimately related to deep questions in combinatorics that have been studied in their own right for well over half a century. Some of these questions have exciting applications in statistics and computer science. In this talk I am going to give an overview of this interdisciplinary research area and of the new contributions that physics insights have enabled.

We consider a family of multivariate autoregressive stochastic sequences that restart when hit a neighbourhood of the origin, and study the distributional limit of their stationary distributions when the autoregressive coefficient tends to one, the noise scaling parameter tends to zero, and the neighbourhood size varies. We obtain a non-standard limit theorem where the limiting distribution is a mixture of an atomic distribution and an absolutely continuous distribution whose marginals, in turn, are mixtures of distributions of signed absolute values of normal random variables. Several examples are discussed.

Mathematical models based on systems of reaction-diffusion equations provide fundamental tools for the description and investigation of various processes in biology, biochemistry, and chemistry; in specific situations, an appealing characteristic of the arising nonlinear partial differential equations is the formation of patterns, reminiscent of those found in nature. The deterministic Gray-Scott equations constitute an elementary two-component system that describes autocatalytic reaction processes; depending on the choice of the specific parameters, complex patterns of spirals, waves, stripes, or spots appear. In the derivation of a macroscopic model such as the deterministic Gray-Scott equations from basic physical principles, certain aspects of microscopic dynamics, e.g.~fluctuations of molecules, are disregarded; an expedient mathematical approach that accounts for significant microscopic effects relies on the incorporation of stochastic processes and the consideration of stochastic partial differential equations. The randomness leads to a variate of new phenomena and may have a highly non-trivial impact on the behaviour of the solution. E.g. it has been shown by numerical modelling that the stochastic extension leads to a broader range of parameter with Turing patterns by a genetically engineered synthetic bacterial population in which the signalling molecules form a stochastic activator-inhibitor system. The stochastic extension may lead to multistability and noise-induced transitions between different states. In the talk, we will introduce the Gray Scott system, which is a special case of an activator-inhibitor system. Then, we introduce its numerical modelling and highlight the proof of convergence. References: Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray-Scott equations. J. Comput. Appl. Math. 364, Joint work with Jonas Toelle: A Schauder Tychonoff type Theorem and the stochastic Klausmeier system (Archive)

The Kardar-Parisi-Zhang (KPZ) universality class of stochastic growth models has several connections with other a-priori unrelated models in mathematics physics. We will describe which models belongs to the KPZ class, present some of the known result, and highlight some connections, in particular the one with random polymers and random matrices.

We consider large non-Hermitian real or complex random matrices X with independent, identically distributed centred entries. We prove that their local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of the Ginibre ensemble, i.e. when the matrix elements of X are Gaussian. This result is the non-Hermitian counterpart of the universality of the Tracy-Widom distribution at the spectral edges of the Wigner ensemble. This is a joint work with Giorgio Cipolloni and Dominik Schroder.

There is no rigorous mathematical definition of a quasicrystal. In spaces with some group translation the latter term usually refers to well-scattered point sets (Delone sets) that are not periodic but display long range symmetries. Classes of these sets such as model sets have already been studied by Meyer in the 70's, i.e. some time before Shechtman's discovery of physical alloys with non-periodic molecular structure in 1982 (Nobel prize for chemistry in 2011). In this talk we focus on non-periodic point sets in lcsc groups (with the Heisenberg group as a guiding example) that are not too far from crystals in a dynamical sense. The main focus will be on the generalization of the concept of linearly repetitive Delone sets known from Euclidean space. Although they are not found easily, non-periodic, linearly repetitive Delone sets exist in many non-Abelian groups as well. Going beyond a result from Lagarias and Pleasants for R^d roughly outline how to prove unique ergodicity for the associated dynamical systems for a class of Lie groups of polynomial volume growth. Joint work with Siegfried Beckus and Tobias Hartnick.

Dyson's threefold approach suggests to deal with real/complex/quaternion random matrices as beta=1/2/4 instances of beta-ensembles. We complement this approach by the beta=\infty point, whose study reveals a number of previously unnoticed algebraic structures. Our central object is the G\inftyE ensemble, which is a counterpart of the classical Gaussian Orthogonal/Unitary/Symplectic ensembles. We encounter unusual orthogonal polynomials, random walks, and finite free polynomial convolutions.

We investigate large deviation principles for lacunary (trigonometric) sums and show that the large deviations are sensitive to the properties (not only the growth) of the lacunary sequence. No prior knowledge is assumed. Joint work with Christoph Aistleitner, Zakhar Kabluchko, Joscha Prochno und Kavita Ramanan.

Compressive Sensing is a recent development in mathematical signal processing which predicts that vectors that are approximately sparse can be accurately reconstructed from a number of linear measurements that is much smaller than previously believed possible. Efficient algorithms such as convex optimization approaches can be used for the reconstruction. Somewhat surprisingly random matrices model provably optimal measurement schemes. While the most accurate analysis of this phenomenon is available for Gaussian random matrices, practical applications demand for more structure in the measurement matrix. Structured random matrices of particular interest arise from randomly sampling the Fourier transform of the signal as well as subsampling the convolution of the signal with a random vector. We will discuss tools for analyzing the latter type of structured random matrices. It turns out that this leads to bounding the supremum of certain second order chaos processes. In particular, we will present a recent result that provides generic chaining type bounds via gamma-2-functionals. Joint work with Felix Krahmer and Shahar Mendelson.