T8: Random discrete structures and their limits

Discrete structures are omnipresent in mathematics, computer science, statistical physics, optimisation and models of natural phenomena. For instance, complex random graphs serve as a model for social networks or the world wide web. Such structures can be descriptions of objects that are intrinsically discrete or they occur as an approximation of continuous objects. An intriguing feature of random discrete structures is that the models exhibit complex macroscopic behaviour, phase transitions in a wide sense, making the field a rich source of challenging mathematical questions. In this topic we will concentrate on three strands of random discrete structures that combine various research interests and expertise present in Münster.

  • Mathematical fields

    • Model theory and set theory
    • Operator algebras and mathematical physics
    • Applied analysis and theory of PDEs
    • Stochastic analysis
    • Theory of stochastic processes

  • Collaborations with other Topics

  • Selected publications and preprints

    since 2019

    $\bullet $ Chiranjib Mukherjee and Konstantin Recke. Schur multipliers of C$^*$ algebras, group-invariant compactification and applications to amenability and percolation. Journal of Functional Analysis, 287(2):Paper No. 110468, July 2024. doi:10.1016/j.jfa.2024.110468.

    $\bullet $ Florian Besau, Anna Gusakova, Matthias Reitzner, Carsten Schütt, Christoph Thäle, and Elisabeth M. Werner. Spherical convex hull of random points on a wedge. Mathematische Annalen, 389(3):2289–2316, August 2023. doi:10.1007/s00208-023-02704-9.

    $\bullet $ Brian C. Hall, Ching-Wei Ho, Jonas Jalowy, and Zakhar Kabluchko. Zeros of random polynomials undergoing the heat flow. arXiv e-prints, August 2023. arXiv:2308.11685.

    $\bullet $ Rodrigo Bazaes, Chiranjib Mukherjee, Alejandro F. Ramírez, and Santiago Saglietti. Quenched and averaged large deviations for random walks in random environments: The impact of disorder. The Annals of Applied Probability, 33(3):2210–2246, June 2023. doi:10.1214/22-AAP1864.

    $\bullet $ Matthias Erbar, Martin Huesmann, Jonas Jalowy, and Bastian Müller. Optimal transport of stationary point processes: Metric structure, gradient flow and convexity of the specific entropy. arXiv e-prints, April 2023. arXiv:2304.11145.

    $\bullet $ Chiranjib Mukherjee and Konstantin Recke. Haagerup property and group-invariant percolation. arXiv e-prints, March 2023. arXiv:2303.17429.

    $\bullet $ Stephan Luckhaus and Angela Stevens. A two level contagion process and its deterministic McKendrick limit with relevance for the Covid epidemic. Ensaios Matemáticos, 2023. doi:10.21711/217504322023/em3813.

    $\bullet $ Zakhar Kabluchko, Matthias Löwe, and Kristina Schubert. Fluctuations of the magnetization for ising models on Erdős-Rényi random graphs – the regimes of low temperature and external magnetic field. Lat. Am. J. Probab. Math. Stat, 19:537––563, August 2022. doi:10.30757/ALEA.v19-21.

    $\bullet $ Zakhar Kabluchko. Lee-Yang zeroes of the Curie-Weiss ferromagnet, unitary Hermite polynomials, and the backward heat flow. arXiv e-prints, March 2022. arXiv:2203.05533.

    $\bullet $ Zakhar Kabluchko. Angles of random simplices and face numbers of random polytopes. Adv. Math., March 2021. doi:10.1016/j.aim.2021.107612.

    $\bullet $ Matthias Erbar, Martin Huesmann, and Thomas Leblé. The one-dimensional log-gas free energy has a unique minimizer. Commun. Pure Appl. Math., 74(3):615–675, January 2021. doi:10.1002/cpa.21977.

    $\bullet $ Zakhar Kabluchko, Matthias Löwe, and Kristina Schubert. Fluctuations of the magnetization for Ising models on Erdős-Rényi random graphs—the regimes of small $p$ and the critical temperature. J. Phys. A, 53(35):355004, 37, August 2020. doi:10.1088/1751-8121/aba05f.

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  • Recent publications and preprints

    since 2023

    $\bullet $ Chiranjib Mukherjee and Konstantin Recke. Schur multipliers of C$^*$ algebras, group-invariant compactification and applications to amenability and percolation. Journal of Functional Analysis, 287(2):Paper No. 110468, July 2024. doi:10.1016/j.jfa.2024.110468.

    $\bullet $ David Dereudre, Daniela Flimmel, Martin Huesmann, and Thomas Leblé. (non)-hyperuniformity of perturbed lattices. arXiv e-prints, May 2024. arXiv:2405.19881.

    $\bullet $ Anna Gusakova, Zakhar Kabluchko, and Christoph Thäle. Sectional Voronoi tessellations: characterization and high-dimensional limits. Bernoulli, May 2024. doi:10.3150/23-bej1641.

    $\bullet $ Martin Huesmann and Thomas Leblé. The link between hyperuniformity, Coulomb energy, and Wasserstein distance to Lebesgue for two-dimensional point processes. arXiv e-prints, April 2024. arXiv:2404.18588.

    $\bullet $ Martin Brückerhoff and Martin Huesmann. Shadows and barriers. The Annals of Applied Probability, February 2024. doi:10.1214/23-aap1981.

    $\bullet $ Martin Huesmann and Bastian Müller. A Benamou-Brenier formula for transport distances between stationary random measures. arXiv e-prints, February 2024. arXiv:2402.04842.

    $\bullet $ Martin Huesmann, Francesco Mattesini, and Felix Otto. There is no stationary p-cyclically monotone Poisson matching in 2d. Electron. J. Probab., January 2024. doi:10.1214/24-ejp1171.

    $\bullet $ Matthias Löwe and Sara Terveer. Spectral properties of the strongly assortative stochastic block model and their application to hitting times of random walks. arXiv e-prints, January 2024. arXiv:2401.07896.

    $\bullet $ Michael Goldman, Martin Huesmann, and Felix Otto. Almost sharp rates of convergence for the average cost and displacement in the optimal matching problem. arXiv e-prints, December 2023. arXiv:2312.07995.

    $\bullet $ Martin Huesmann, Francesco Mattesini, and Dario Trevisan. Wasserstein asymptotics for the empirical measure of fractional Brownian motion on a flat torus. Stochastic Processes and their Applications, 155:1–26, October 2023. doi:10.1016/j.spa.2022.09.008.

    $\bullet $ Martin Huesmann, Francesco Mattesini, and Felix Otto. There is no stationary cyclically monotone Poisson matching in 2d. Probability Theory and Related Fields, 187(3-4):629–656, August 2023. doi:10.1007/s00440-023-01225-5.

    $\bullet $ Florian Besau, Anna Gusakova, Matthias Reitzner, Carsten Schütt, Christoph Thäle, and Elisabeth M. Werner. Spherical convex hull of random points on a wedge. Mathematische Annalen, 389(3):2289–2316, August 2023. doi:10.1007/s00208-023-02704-9.

    $\bullet $ Brian C. Hall, Ching-Wei Ho, Jonas Jalowy, and Zakhar Kabluchko. Zeros of random polynomials undergoing the heat flow. arXiv e-prints, August 2023. arXiv:2308.11685.

    $\bullet $ Rodrigo Bazaes, Chiranjib Mukherjee, Alejandro F. Ramírez, and Santiago Saglietti. Quenched and averaged large deviations for random walks in random environments: The impact of disorder. The Annals of Applied Probability, 33(3):2210–2246, June 2023. doi:10.1214/22-AAP1864.

    $\bullet $ Brian Hall, Ching-Wei Ho, Jonas Jalowy, and Zakhar Kabluchko. The heat flow, GAF, and SL(2;R). arXiv e-prints, April 2023. doi:2304.06665.

    $\bullet $ Matthias Erbar, Martin Huesmann, Jonas Jalowy, and Bastian Müller. Optimal transport of stationary point processes: Metric structure, gradient flow and convexity of the specific entropy. arXiv e-prints, April 2023. arXiv:2304.11145.

    $\bullet $ Chiranjib Mukherjee and Konstantin Recke. Haagerup property and group-invariant percolation. arXiv e-prints, March 2023. arXiv:2303.17429.

    $\bullet $ Martin Huesmann and Bastian Müller. Transportation of random measures not charging small sets. arXiv e-prints, March 2023. arXiv:2303.00504.

    $\bullet $ Jonas Jalowy, Zakhar Kabluchko, Matthias Löwe, and Alexander Marynych. When does the chaos in the Curie-Weiss model stop to propagate? Electronic Journal of Probability, January 2023. doi:10.1214/23-ejp1039.

    $\bullet $ Stephan Luckhaus and Angela Stevens. A two level contagion process and its deterministic McKendrick limit with relevance for the Covid epidemic. Ensaios Matemáticos, 2023. doi:10.21711/217504322023/em3813.

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    further publications

Back to research programme

Random tessellations and polytopes

© MM/vl

Investigators: Gusakova, Huesmann, Kabluchko

One of the research direction in random discrete structures is random polytopes and tessellations an important area in stochastic geometry. Two of our main objects of interest are random beta polytopes and Voronoi and Laguerre tessellations. In the former we consider the convex hull of points sampled from a beta distribution, determine various relevant functionals such as the expected number of faces and study threshold phenomena as the number of points and the dimension go to infinity. In the latter we analyse the influence of attractive or repulsive forces on the point process to the geometry of the tessellations.

Random graphs and percolation

Investigators: Dereich, Geffen, Jalowy, Kabluchko, Kerr, Löwe, Mukherjee, Tent

The unit on random graphs and percolation opens up another aspect of random geometries. Our interest ranges from classical Erdős-Rényi graphs to preferential attachment models. We use random graphs as an environment for spin systems and address the relation of ER graphs and percolation. We investigate percolation on groups to examine their properties as well as those of the related operator algebras.

Limits of large particle systems

Investigators: Huesmann, Jalowy, Kabluchko, Mukherjee, Stevens

In this unit we investigate specific interacting particle systems and their limits. We employ large deviation principles for random walks in random environments to study the impact of disorder on stochastic homogenisation of the Hamilton–Jacobi–Bellman equation and use regularity structures to prove convergence of WASEP to the KPZ equation and convergence of static Ising–Kac models to $\Phi^4_3$-theory.