#### Publications and Preprints

**Cokernel statistics for walk matrices of directed and weighted random graphs**

[arXiv]

## ▷Abstract

The walk matrix associated to an n×n integer matrix X and an integer vector b is defined by W:=(b,Xb,…,Xn−1b). We study limiting laws for the cokernel of W in the scenario where X is a random matrix with independent entries and b is deterministic. Our first main result provides a formula for the distribution of the pm-torsion part of the cokernel, as a group, when X has independent entries from a specific distribution. The second main result relaxes the distributional assumption and concerns the Z[x]-module structure.

The motivation for this work arises from an open problem in spectral graph theory which asks to show that random graphs are often determined up to isomorphism by their (generalized) spectrum. Sufficient conditions for generalized spectral determinacy can namely be stated in terms of the cokernel of a walk matrix. Extensions of our results could potentially be used to determine how often those conditions are satisfied. Some remaining challenges for such extensions are outlined in the paper.

**Matrix concentration inequalities with dependent summands and sharp leading-order terms**With Jaron Sanders.

[arXiv] [YouTube]

## ▷Abstract

We establish sharp concentration inequalities for sums of dependent random matrices. Our results concern two models. First, a model where summands are generated by a ψ-mixing Markov chain. Second, a model where summands are expressed as deterministic matrices multiplied by scalar random variables. In both models, the leading-order term is provided by free probability theory. This leading-order term is often asymptotically sharp and, in particular, does not suffer from the logarithmic dimensional dependence which is present in previous results such as the matrix Khintchine inequality.

A key challenge in the proof is that techniques based on classical cumulants, which can be used in a setting with independent summands, fail to produce efficient estimates in the Markovian model. Our approach is instead based on Boolean cumulants and a change-of-measure argument.

We discuss applications concerning community detection in Markov chains, random matrices with heavy-tailed entries, and the analysis of random graphs with dependent edges.

**Detection and evaluation of clusters within sequential data.**With Albert Senen-Cerda, Gianluca Kosmella, and Jaron Sanders.

[arXiv]

## ▷Abstract

Motivated by theoretical advancements in dimensionality reduction techniques we use a recent model, called Block Markov Chains, to conduct a practical study of clustering in real-world sequential data. Clustering algorithms for Block Markov Chains possess theoretical optimality guarantees and can be deployed in sparse data regimes. Despite these favorable theoretical properties, a thorough evaluation of these algorithms in realistic settings has been lacking.

We address this issue and investigate the suitability of these clustering algorithms in exploratory data analysis of real-world sequential data. In particular, our sequential data is derived from human DNA, written text, animal movement data and financial markets. In order to evaluate the determined clusters, and the associated Block Markov Chain model, we further develop a set of evaluation tools. These tools include benchmarking, spectral noise analysis and statistical model selection tools. An efficient implementation of the clustering algorithm and the new evaluation tools is made available together with this paper.

Practical challenges associated to real-world data are encountered and discussed. It is ultimately found that the Block Markov Chain model assumption, together with the tools developed here, can indeed produce meaningful insights in exploratory data analyses despite the complexity and sparsity of real-world data.

**Singular value distribution of dense random matrices with block Markovian dependence.**With Jaron Sanders.

*Stochastic Processes and their Applications*(2023).

[arXiv] [Journal]

## ▷Abstract

A block Markov chain is a Markov chain whose state space can be partitioned into a finite number of clusters such that the transition probabilities only depend on the clusters. Block Markov chains thus serve as a model for Markov chains with communities. This paper establishes limiting laws for the singular value distributions of the empirical transition matrix and empirical frequency matrix associated to a sample path of the block Markov chain whenever the length of the sample path is Θ(n²) with n the size of the state space.

The proof approach is split into two parts. First, we introduce a class of symmetric random matrices with dependence called approximately uncorrelated random matrices with variance profile. We establish their limiting eigenvalue distributions by means of the moment method. Second, we develop a coupling argument to show that this general-purpose result applies to block Markov chains.

**Estimates for zero loci of Bernstein-Sato ideals.**With Nero Budur and Robin van der Veer.

*To appear in Publications of the Research Institute for Mathematical Sciences*(2023+).

[arXiv] [Journal]

## ▷Abstract

We give estimates for the zero loci of Bernstein-Sato ideals. An upper bound is proved as a multivariate generalisation of the upper bound by Lichtin for the roots of Bernstein-Sato polynomials. The lower bounds generalise the fact that log-canonical thresholds, small jumping numbers of multiplier ideals, and their real versions provide roots of Bernstein-Sato polynomials.

#### Talks, posters, and more

**Talk: Cokernel statistics of walk matrices: a random matrix approach towards generalized spectral determinacy of random graphs**at 6th Dutch-Belgian discrete mathematics seminar. (2024) [slides]

**Talk: Cokernel statistics of walk matrices: Towards generalized spectral determinacy of random graphs**at TU/e lunch seminar in Combinatorial Optimization. (2024) [slides]

**Talk: Matrix concentration inequalities with dependent summands and sharp leading-order terms**at SNAPP Seminar. (2023) [slides] [YouTube]

**Discussion Paper Contribution: ‘Vintage Factor Analysis with Varimax Performs Statistical Inference’ by Rohe & Zeng**published at Journal of the Royal Statistical Society Series B: Statistical Methodology. (2023) [Journal]

**Talk: Matrix concentration inequalities with dependent summands and sharp leading-order terms**at KU Leuven Classical Analysis Seminar. (2023) [slides]

**Talk: Detection and evaluation of clusters within animal movements**at Stochastic Models in Life Science (Eurandom). (2023) [slides]

**Poster: Universality-based concentration for sums of dependent random matrices**at 21st INFORMS Applied Probability Society Conference (APS Nancy). (2023) [pdf] [Best poster award]

**Talk: Detection and evaluation of clusters within sequential data**at 21st INFORMS Applied Probability Society Conference (APS Nancy). (2023) [slides]

**Talk: Matrix concentration inequalities with dependent summands and sharp leading-order terms**at 10th High Dimensional Probability conference (Będlewo). (2023) [abstract] [slides]

**Talk: Universality-based concentration for sums of dependent random matrices**at 9th SOR PhD Colloquium. (2023) [abstract]

**Poster: Universality-based concentration for sums of dependent random matrices**at 18th Young European Probabilists workshop (YEP XVIII). (2023)

**Talk: Clusters within Markov chains: Detection, evaluation, and spectral fingerprints**at 14th NETWORKS training week. (2022) [slides]

**Talk: Bulk of random matrices generated by Markov chains with community structure**at fourth ZiF summer school on randomness in physics and mathematics. (2022) [slides]

**Talk: Spectra of random matrices with Markovian dependence and non-constant variance profile**at 50th Saint-Flour probability summer school. (2022) [abstract]

**Poster: Singular value distribution of random matrices with block Markovian dependence**at XVII Brunel-Bielefeld Workshop. (2021) [pdf]

#### Figures

They say a figure speaks a thousand words. The following figures are taken from the papers and preprints above.