Program

Title: Rigidity, hyperuniformity and related phenomena: theory and applications 

Some recent large deviations results for Coulomb gases and random matrices

Recent advances in the Riesz gas: statics and dynamics

Title: Real eigenvalues of asymmetric random matrices

Abstract: In the study of non-Hermitian random matrix theory, a unique feature arising in the class of orthogonal symmetry is the non-trivial probability of having purely real eigenvalues. In this talk, focusing on real Ginibre matrices and their invariants, particularly the elliptic Ginibre ensembles, I will discuss how fundamental concepts in probability theory—the law of large numbers, the central limit theorem, and large deviations—are developed in the study of real eigenvalues of asymmetric random matrices. 

Title: Massive Particle Systems, Wasserstein Brownian Motions, and the Dean--Kawasaki SPDE 

Let W be a conservative, ergodic Markov diffusion on some arbitrary state space M, converging exponentially fast to equilibrium. We consider:

(1) Systems of up to countably many massive particles in M, with finite total mass. Each particle is subject to an independent instance of the noise W, with volatility the inverse mass carried by the particle. We prove that the corresponding infinite system of SDEs has a unique solution, for every starting configuration and every distribution of the masses in the infinite simplex.

(2) Solutions to the Dean–Kawasaki SPDE with singular drift, driven by the generator L of W. We prove that the equation may be given rigorous meaning on M, and that it has a unique `distributional’ solution. This extends Konarovskyi–Lehmann–von Renesse's `ill-posedness vs. triviality' to the case of infinitely many massive particles.

(3) Diffusions with values in the space P of all probability measures on M, driven by the geometry induced by L.

(4) In the case when M is a manifold, differential-geometric and metric-measure Brownian motions on P induced by the geometry of optimal transportation and reversible for a normalized completely random measure.

We show that all these objects coincide.

Title: (Non-)hyperuniformity of perturbed lattices 

Abstract: By a lattice, we understand a subgroup of the $\mathbb{R}^d$ generated by linear combinations with integer coefficients of the vectors of a certain basis of $\R^d$. The basic example is the so-called integer lattice $\mathbb{Z}^d$. By uniformly shifting the whole lattice and by displacing (i.e. perturbing) each individual point of the lattice according to some probability law, we obtain a very manageable stationary point process - a perturbed lattice. It has been shown recently that this creates a rather versatile tool. Depending on the dimension and the covariance structure among the perturbations, perturbing a lattice allows to generate a point process with arbitrary order of the density fluctuations - from class I hyperuniform to hyperfluctuating point patterns. 

 METASTABILITY FOR THE CURIE–WEISS–POTTS MODEL WITH UNBOUNDED RANDOM INTERACTIONS

Abstract: 

I will first introduce the model, i.e. a disordered version of the mean-field q-spin Potts model (CWP), where the interaction coefficients between spins are general independent random variables. These random variables are chosen to have fixed mean (for simplicity taken to be 1), well defined log-moment generating function and finite variance.

I will then present quantitative estimates of metastability in the regime of large number of particles at fixed temperature, when the system evolves according to a Glauber dynamics. This means that the spin configuration is viewed as a Markov chain where spins flip according to Metropolis rates at a fixed inverse temperature. Our main result identifies conditions ensuring that, with high probability, the system behaves like the corresponding system where the random couplings are replaced by their averages. More precisely, we prove that the metastability of the former system is implied with high probability by the metastability of the latter. Moreover, we consider relevant metastable hitting times of the two systems and find the asymptotic tail behaviour and the moments of their ratio. Our proofs use the potential-theoretic approach to metastability in combination with concentration inequalities. Based on a joint work in collaboration with Johan Dubbeldam, Vicente Lenz and Martin Slowik. 

Title: Stochastic geometry, Poisson processes and Stein's method

Abstract: Stochastic geometry is the branch of probability theory dealing with spatial random structures such as random tessellations, random sets, random polytopes or spatial random graphs. Such objects are often constructed from underlying point samples. In many cases and also throughout this talk, it is assumed that these points are given by a Poisson process. Thus, quantities of interest are random variables depending only on a Poisson process, so-called Poisson functionals. Since random geometric structures and associated random variables usually exhibit an extremely complex behaviour, which does not admit explicit finite size descriptions, one studies the asymptotic behaviour as the number of underlying points tends to infinity. In order to establish central limit theorems for this situation, one is interested in approximating distributions of Poisson functionals by normal distributions. A powerful tool to establish such results is the Malliavin-Stein method, which will be discussed in this talk. It combines Stein's method, a collection of techniques to derive quantitative limit theorems, with Malliavin calculus, a variational calculus for random variables. To illustrate the use of the Malliavin-Stein method, some problems from stochastic geometry will be considered. 

A CLT for a Graph in Hyperbolic Geometry

Abstract: Given an infinite collection of points, connect each point to its nearest neighbour, then its second nearest neighbour and so on, until the point is included in the convex hull of its neighbours. This is called the nearest neighbour embracing graph, and we study its degree counts and total edge length in Euclidean space and in hyperbolic space in parallel. In this talk, we will see how to derive a central limit theorem for these quantities using the Malliavin-Stein method, and how these results differ if we pass from a Euclidean framework to a hyperbolic one. This is joint work with Holger Sambale and Christoph Thäle.

Limiting eigenvalue distribution of heavy-tailed Toeplitz matrices

We show that under an appropriate scaling, the limiting eigenvalue distribution of a symmetric Toeplitz matrix with i.i.d. entries drawn from an \alpha-stable distribution (0 < \alpha < 2) converges weakly to a random symmetric probability distribution on \mathbb{R}. We express this random probability distribution in terms of the spectral measure of a random unbounded operator on \ell^2(\mathbb{Z}) and study some properties of this distribution for different values of \alpha. Based on joint work with Arnab Sen

Optimal transport for hyperuniform point processes

Abstract: In this talk I will present a recent result obtained in collaboration with S.Dallaporta and D.Garcia-Zelada. We show that the average transport cost between a finite sample of a hyperuniform point process and its intensity measure is bounded by a multiple of the number of points. In other words, regarding optimal transport, hyperuniform point processes behave like perturbed lattice. We will also present how the complex relations between hyperuniformity, linear optimal transport and perturbed lattices was explored by different group of researchers in the last months, giving now a (nearly) full picture of the relations between those notions. 

Title: Probabilistic limits of root dynamics for polynomials and entire functions

Abstract: This talk will be a quick introduction into how the evolution of polynomial roots under differential operators can be described using random matrices. Our motivation will be the fairly straight forward question: Given an ``infinite degree"" polynomial (really an entire function) what is the long time behavior of roots under repeated differentiation? We will see how translating this question into probabilistic limit theorems (e.g. the LLN, CLT, and Poisson limit theorem) can provide valuable insights. This is based on joint work with Sean O'Rourke and David Renfrew

Title: Microscopic behavior of a 1+1 dimensional Coulomb gas.

Abstract: We study a 1+1 dimensional Coulomb gas seen as an interpolation between two important models: the Ginibre ensemble, and zeroes of the random Kac polynomials. At the macroscopic level, the points of this process live in 2 dimensions, but they concentrate at the limit in the unit circle. We examine the microscopic behavior of this point process, and express its limit as a determinantal point process, which exhibits remarkable rigidity properties. We also characterize it as the minimizer of a "free energy" functional through a large deviation principle.

Title: Superbosonization for structured random matrices

Abstract:  Imagine an $NK\times NK$ Hermitian matrix with independent Gaussian entries that is divided into $K\times K$ blocks of size $N \times N$. Let each entry in the $(i, j)$ block have mean zero and variance $S_{ij} \geq 0$. Clearly, the distribution of eigenvalues depends on the $K\times K$ variance profile matrix $S$. E.g. in previous work by Torben Krüger and David Renfrew it was shown that the large-$N$ limit of the eigenvalue density can develop a singularity at the origin and the singularity degree depends on the pattern of zeros in $S$.\\

I will present results from ongoing work with Torben Krüger where we determine the eigenvalue density near the origin when $S$ has a staircase structure, i.e. $S$ has non-zero entries only along the anti-diagonal and one line above the anti-diagonal. We derive the density via its Stieltjes-transform, for which we compute an exact formula at finite-$N$ using the superbosonization technique. Our method can be extended to $k$-point correlation functions for arbitrary $k$.

Title: Nonhyperuniformity and the inverse Henderson problem

We give a brief introduction to the inverse Henderson problem from statistical mechanics, which deals with the following problem: Given the pair correlation function of some Gibbs measure, determine the interaction. We show how to write this as a minimization problem of a relative entropy functional and highlight the question of a nonhyperuniformity type property appearing in the minimization.

Multidimensional compound Poisson approximation for the Gilbert graph

Fully-connected continuum percolation

Title: Real eigenvalues of the real Ginibre matrices.

Abstract: The real Ginibre matrix ensemble is a fundamental random matrix model characterized by a Pfaffian point process. In this talk, we explore its one-parameter generalization, the real elliptic Ginibre ensemble, focusing on the statistical properties of its real eigenvalues. In particular, we examine the finite-size corrections to the 1-point functions and the upper tail distribution of the real eigenvalue number. These statistical properties exhibit distinct behaviors depending on the parameter of the ensemble. This talk is based on joint works with Gernot Akemann and Sung-Soo Byun.

Abstract: A point process is hyperuniform if the variance of its number of points within a large Euclidean ball scales slower than its volume. Initially conceptualized in statistical physics, hyperuniformity appears in various applied contexts. Despite this, statistical tests hyperuniformity have only recently attracted attention. The notion of hyperuniformity can be redefined using the structure factor S. More precisely, under certain conditions, hyperuniformity becomes equivalent to the structure factor S(k) vanishing at zero frequency k=0. A common assumption is that S follows a power-law behavior near zero, whose exponent is called the hyperuniformity exponent. In this pitch-talk, we present an estimator of the hyperuniformity exponent and discuss its theoretical properties (convergence, asymptotic confidence intervals).

Sequence matching is a fundamental problem in data analysis with applications spanning computer vision, speech processing, bioinformatics, and natural language processing. However, efficiently solving these problems is challenging due to the need to maintain temporal consistency, preserve neighborhood structure similarity, while also allowing flexibility in start-end matching points. Although dynamic time warping (DTW) obtains a global optimal solution, it does not necessarily achieve locally sensible matchings. The Wasserstein distance, particularly when used in the context of Optimal Transport (OT), can accommodate local inversions of the order of elements and differences in starting points of sequences. This provides a more intuitive and flexible matching. Moreover the Wasserstein distance allows the integration of multiple features, such as the similarity of elements and their temporal positions. To address the issue of mismatching points with distinct neighborhood structures, we incorporate local structure descriptors that capture point-wise structural information. We introduce three novel OT-based dissimilarity measures: (1) Bag-of-Pattern Wasserstein, inspired by edit-distance and bag-of-pattern approaches, compares the histogram representations for time series data; (2) ShapeWasserstein inspired by shapeDTW, which takes advantage of the flexibility of Wasserstein matching; and (3) Co-Optimal Transport for graph matching, which compares the structural properties of digraphs obtained by transforming time series data. Additionally, to enhance classification performance, we integrate different measures as weak learners into an ensemble model using stacked generalization techniques.

Keywords: Dynamic time warping, Optimal transport, global patterns and local patterns recognization, clustering method, stacked generalisation, Graph represenation of time series

Interactions between different birds of prey as a random point process

The two-dimensional (2D) Coulomb gas is a one-parameter family of random point processes, depending on the inverse temperature β. Based on previous work, it is proposed as a simple statistical measure to quantify the intra- and interspecies repulsion among three different highly territorial birds of prey. Using data from the area of the Teutoburger Wald over 20 years, we fit the nearest-neighbour and next-to-nearest neighbour spacing distributions between the respective nests of the goshawk, eagle owl and the previously examined common buzzard to β of the Coulomb gas.

Visibility in Brownian interlacements, Poisson cylinders and Boolean models

Title: Emergence of a Poisson process in weakly interacting particle systems

Abstract: This talk will be about an interacting particle system driven by two forces: a repulsive pairwise interaction between them, and a confining potential. At a positive temperature, their behavior is driven by the Gibbs measure associated with this Hamiltoninan.In this talk, we will be interested in a certain class of “weakly interacting ” pairwise interactions. For this kind of interaction, we will show that the local behavior of the system is asymptotically given by a Poisson Point Process. We show this under more general assumptions on the temperature scaling than previous works in the literature. 

Title: Large deviation principle for binomial Gibbs processes

Abstract: 

Gibbs processes in the continuum are one of the most fundamental models in spatial stochastics. They are typically defined using a density with respect to the Poisson point process. In the language of statistical mechanics, this corresponds to the grand-canonical ensemble, where the number of particles is random. Of the same importance is the canonical ensemble, where the number of particles is fixed. In the language of point processes, this corresponds to studying binomial Gibbs processes which are defined using a density with respect to the binomial point process. 

In this talk, we present a large deviation theory developed for functionals of binomial Gibbs processes with fixed intensity in increasing windows. Our method relies on the traditional large deviation result from [1] noting that the binomial point process is obtained from the Poisson point process by conditioning on the point number. Our main methodological contribution is the development of coupling constructions allowing us to handle  delicate and unlikely pathological events. The presented results cover a broad class of both the interaction function (possibly unbounded) and the functionals (given as a sum of possibly unbounded local score functions). 

[1] Georgii, H.-O. and Zessin, H. (1993): Large deviations and the maximum entropy principle for marked point random fields, Probab. Theory Related Fields 96, 177--204.

Title : Liquid-gas phase transition for Gibbs point processes with saturated interaction

Abstract : Gibbs point processes are natural objects to study systems of particles in interaction. In finite volume, the unnormalised density of the Gibbs measure with respect to a Poisson point process with activity $z$ is given by the Boltzmann factor $e^{-\beta H}$, where $\beta$ is the inverse temperature and $H$ is the Hamiltonian that encodes the interaction between particles. The infinite volume Gibbs point processes are defined as solutions to the Dobrushin-Lanford-Ruelle equations, which describe the equilibrium of the system. A liquid-gas phase transition occurs when we do not have uniqueness of the infinite volume Gibbs point process and that they have different densities. We explore the occurrence of such phenomenon in the context of saturated interactions. These interactions represent a class of models where the energy cost of adding a point in areas of high particle density is constant. The Quermass interaction exhibits such saturation property. We will present another interesting example: the diluted pairwise interaction. Under some assumptions, we prove the existence of liquid-gas phase transition for saturated interactions using an adaptation of Pirogov-Sinaï-Zahradník theory in the continuous setting

Poisson Cylinder Process

Dirichlet processes for sequential decision making under uncertainty

First contact percolation (a spin-off of first passage percolation)

The so called first passage percolation model on a graph is a highly intriguing model of describing the spread of e.g. an infection. In the original model, the infection spreads from the origin along edges to other vertices. However, each such passage only happens after some ""waiting time"" attached to each edge.

We want to take a different approach: What if the spread is not determined by some ""waiting time"" but rather whether there is a contact between two vertices? These contact times are represented by point processes and we call the resulting model ""first contact percolation"

Metric based approaches for spatial random graphs

In the field of spatial point processes, the OSPA metric and the TT metric are two intuitive metrics between point patterns of (possibly) different sizes that are based on optimal assignments. Previous research showed their suitability for statistical analysis and how they may be employed to investigate weak convergence of point processes via Stein's method. 

Building upon these two point pattern metrics, we introduce a metric between spatial graphs of (possibly) different sizes that, additional to vertex differences, includes the discrepancy of edge structures in the optimal assignment. Using a representation of a graph as a pair of point processes we derive a new type of (random) graph convergence and study this convergence by developing Stein's method and deriving convergence rates. Finally, we present an application of these convergence results to thinned spatial random graphs.

Joint work with Dominic Schuhmacher (University of Göttingen).

Large Deviations for Inhomogeneous Random Graphs

We will discuss the double phase transition appearing in non-linear large deviations for subgraph countings in random graphs within the context of Graph Limit Theory. We conclude with some progress towards a large deviation principle for spatial random graphs.