Publications and Preprints
Singular value distribution of dense random matrices with block Markovian dependence. With Jaron Sanders. [arXiv]▷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.
▷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.
Conference contributions
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) [pdf] Poster: Singular value distribution of random matrices with block Markovian dependence at XVII Brunel-Bielefeld Workshop. (2021) [poster]Figures
They say a figure speaks a thousand words. The following figures are taken from the papers and preprints above.
Visualization of a block Markov chain with K = 3 clusters.
Taken from “Singular value distribution of dense random matrices with block Markovian dependence.” with Jaron Sanders.
Visualization of a coupling argument which played a key role in my paper establishing the singular value distributions associated to block Markov chains. The coupling argument allows one to deal with the dependence intrinsic to a Markov chain.
Taken from “Singular value distribution of dense random matrices with block Markovian dependence.” with Jaron Sanders.
Singular value distribution of the empirical transition matrix associated with a block Markov chain.
Taken from “Singular value distribution of dense random matrices with block Markovian dependence.” with Jaron Sanders.
The zero locus of the Bernstein-Sato ideal associated with f_1 = y^2 – x^2 + x^3 and f_2 = y.
Taken from “Estimates for zero loci of Bernstein-Sato ideals.” with Nero Budur and Robin van der Veer. Computed using the library dmodideal.lib from SINGULAR.