Research areas

Complex networks

• scale-free networks

• spatially extendend networks

• directed random graphs

• stochastic processes on (random) graphs

A few projects:

Inhomogenous long-range networks: connectivity & dynamics. This project is a continuation of my work within the DFG-Priority Programme SPP 2265 Random Geometric Systems. The aim is to investigate the properties of infinite clusters in long-range percolation models with dependencies. This includes their topological properties, network metrics such as clustering coefficients etc. and the behaviour of stochastic process on them, for instance random walks or the contact process.

Structure and Dynamics of directed scale-free spatial networks (with Lukas Lüchtrath (WIAS Berlin)). The goal is to investigate directed scale-free network models and stochastic processes on such networks. Considering directed networks, instead of undirected ones, adds additional layers of complexity to the models. The description of networks becomes considerably more involved, even locally, due to the appearance of arbitrary indegree-outdegree correlations. More importantly, the dynamics on directed networks are inherently irreversible, which renders many tools commonly used for the analysis of processes on networks ineffective. Therefore, mathematical results for directed networks are scarce and the effects emerging from introducing directed edges are, in general, poorly understood.

Resolute voter model (with Lisa Hartung (JGU Mainz)). We study a variation of the classical voter model. The voters sit in the sites of a (large) graph and copy the opinion of a random neighbouring voter whenever their clock rings. However, unlike in the classical voter model, the distribution of the clock process depends on the voter: The rate at which a clock rings is itself the inverse of a heavy tailed random variable, i.e. there is an inhomogeneous population of irresolute voters changing their opinions all the time, and resolute voters who change their opinions very rarely. We investigate what effect this inhomogeneity has on the fixation of the system.

Persistence probabilities

• fractional Brownian motion

• self-similar processes

• long-range dependence

This area is concerned with questions of the following type: given a stochastic process whose range almost surely covers the real line, how rare is the event that it remains below a fixed threshold up to time T? For Markov processes the answer is classical, but for non-Markovian processes — in particular those with long-range dependence — the picture is far from complete. For an introduction to the field see this survey by Frank Aurzada & Thomas Simon.

My work on this topic departs from the standard fluctuation-theoretic approach. Using Palm calculus and the analysis of self-similar co-ascent processes, I established universality results for the persistence exponents of local times of self-similar processes with stationary increments: the exponent depends only on the self-similarity index, not on the fine structure of the process. A companion paper develops the Palm-calculus framework underlying these results.

Open question: strong asymptotics for fractional Brownian motion. Molchan's theorem gives the logarithmic order of the persistence probability P(T) of fractional Brownian motion with Hurst index H: log P(T)/log T → -(1−H). The finer question — does P(T) behave like a pure power function? — remains open for general H. I am actively working on this.

Interacting particle systems on random graphs

• infection dynamics

• random walk loop and trace models

• voter models

A central theme in modern probability is understanding how the geometry of a random network shapes the behaviour of dynamical processes running on it. Phase transitions, critical phenomena, and metastability can look very different on sparse random graphs than on regular lattices, and many classical tools break down.

Contact processes on random networks (with Benedikt Jahnel (Braunschweig/WIAS Berlin) and Lukas Lüchtrath (WIAS Berlin)). The contact process is a simple model for the spread of an infection through a population. We study phase transitions and metastability for the contact process on sparse random graphs, exploiting local weak limits and the theory of metastable systems to transfer results from tree-like local structures to the global graph.

Activated random walks and self-organised criticality (with Antal A. Járai (Bath) and Lorenzo Taggi (Rome)). Activated random walk is a conservative particle system that exhibits self-organised criticality: without any parameter tuning, the system naturally settles near a critical point. We investigate the critical window and scaling limits in the mean-field (complete graph) setting.

Strict inequalities for percolation thresholds (with Stein Andreas Bethuelsen (Bergen) and with Andreas Klippel (Darmstadt) & Ben Lees (Leeds)). A recurring question in percolation theory is whether distinct models — say, loop percolation and Bernoulli percolation on a tree, or long-range percolation models with different kernels — have strictly ordered critical values, or whether they can coincide. We develop methods to establish such strict inequalities without relying on classical essential-enhancement arguments.

Preferential attachment and random recursive trees

• preferential attachment graphs

• random recursive trees

• depth and distance asymptotics

• scale-free networks

Preferential attachment models (where new nodes connect preferentially to already well-connected nodes) are a canonical explanation for the emergence of scale-free degree distributions in real-world networks. Despite their simple recursive definition, the geometry of these trees and graphs is subtle and many basic questions about depths, distances, and subtree counts remain open.

My earlier work established the typical and large-deviation behaviour of distances in spatial and non-spatial preferential attachment models. More recently I have returned to the area, studying monotonicity of depth constants in general preferential attachment trees: as the attachment rule becomes more concentrated, do typical depths increase or decrease? The answer turns out to depend delicately on the shape of the attachment function, and proving monotonicity requires developing new comparison techniques for recursive distributional equations.

Research Collaborations

Frank Aurzada (Darmstadt), Stein Andreas Bethuelsen (Bergen), Steffen Dereich (Münster), Peter Gracar (Leeds), Lisa Hartung (Mainz), Markus Heydenreich (Augsburg), Christian Hirsch (Aarhus), Benedikt Jahnel (Braunschweig/WIAS Berlin), Antal A. Járai (Bath), Andreas Klippel (Darmstadt), Vaios Laschos (WIAS Berlin), Ben Lees (Leeds), Lukas Lüchtrath (WIAS Berlin), Peter Mörters (Cologne), Amr Rizk (Hannover), Lorenzo Taggi (Rome), and Florian Völlering

Publications

Preprints

1. Maximising homomorphism counts between digraphs, with Lukas Lüchtrath. arXiv:2603.18847

2. Age-dependent random connection models with arc reciprocity: clustering and connectivity, with Lukas Lüchtrath. arXiv:2603.16824

3. Monotonicity of depth constants in general preferential attachment trees. arXiv:2602.14741

4. Extremes of the zero-average Gaussian Free Field on random regular graphs, with Lisa Hartung and Andreas Klippel. arXiv:2511.14026

5. Strict monotonicity of critical points in independent long-range percolation models, with Stein Andreas Bethuelsen. arXiv:2510.26314

6. Phase transitions for contact processes on sparse random graphs via metastability and local limits, with Benedikt Jahnel and Lukas Lüchtrath. arXiv:2505.22471

7. Loop vs. Bernoulli percolation on trees: strict inequality of critical values, with Andreas Klippel and Ben Lees. arXiv:2503.03319

8. Phase transitions for contact processes on one-dimensional networks, with Benedikt Jahnel and Lukas Lüchtrath. arXiv:2501.16858

9. Inhomogeneous long-range percolation in the strong decay regime: recurrence in one dimension. arXiv:2408.06918

10. The mean field stubborn voter model, with Lisa Hartung and Florian Völlering. arXiv:2405.08202

11. The critical window in activated random walk on the complete graph, with Antal A. Járai and Lorenzo Taggi. arXiv:2304.10169

Peer reviewed publications in journals

1. Finiteness of the percolation threshold for inhomogeneous long-range models in one dimension, with Peter Gracar and Lukas Lüchtrath, Electronic Journal of Probability 30 (2025), paper no. 134, 29 pp.

2. A very short proof of Sidorenko’s inequality for counts of homomorphism between graphs, with Lukas Lüchtrath, Bulletin of the Australian Mathematical Society 113.1 (2026), pp.10-14.

3. Inhomogeneous long-range percolation in the weak decay regime. Probability Theory and Related Fields 189, 3-4 (2024), pp. 1129–1160. MR4771112

4. Self-similar co-ascent processes and Palm calculus. Stochastic Processes and their Applications 174, (2024) paper no. 104378, 10 pp. MR4746578

5. DAG-type Distributed Ledgers via Young-age Preferential Attachment, with Amr Rizk. Stochastic Systems 13.3 (2023), pp. 377-397. MR4650338

6. Recurrence versus transience for Weight-dependent Random Connection Models, with Peter Gracar, Markus Heydenreich, and Peter Mörters. Electronic Journal of Probability 27 (2022), paper no. 60, 31 pp. MR4417198

7. Universality for persistence exponents of local times of self-similar processes with stationary increments. Journal of Theoretical Probability, 35 (2022), pp. 1842–1862. MR4488560

8. Quenched invariance principle for random walks on dynamically averaging random conductances, with Stein Andreas Bethuelsen and Christian Hirsch. Electronic Communications in Probability 26 (2021), paper no. 69, 13 pp. MR4346873

9. Distances and large deviations in the spatial preferential attachment model, with Christian Hirsch. Bernoulli 26.2 (2020), pp. 927--947. MR4058356

10. Persistence probabilities and a decorrelation inequality for the Rosenblatt process and Hermite processes, with Frank Aurzada. Теория вероятностей и ее применения 63 (2018), pp. 817-826 and Theory of Probability and its Applications 63.4 (2019), pp. 664-670. MR3869634

11. Distances in scale-free networks at criticality, with Steffen Dereich and Peter Mörters. Electronic Journal of Probability 22 (2017), paper no. 77, 38 pp. MR3710797

12. Relations between L^p- and pointwise convergence of families of functions indexed by the unit interval, with Vaios Laschos. Real Analysis Exchange, 38.1 (2012/13) pp. 177–192. MR3083205

13. Typical distances in ultrasmall random networks, with Steffen Dereich and Peter Mörters. Advances in Applied Probability, 44.2 (2012), pp. 583–601. MR2977409

Peer reviewed contributions to conferences and book chapters

1. The directed Age-dependent Random Connection Model with arc reciprocity, with Lukas Lüchtrath. Modelling and Mining Networks. 19th International Workshop, WAW 2024, Warsaw, Poland, June 3–6, 2024, Proceedings. Lecture Notes in Computer Science, vol 14671, 2024.

2. The emergence of a giant component in one-dimensional inhomogeneous networks with long-range effects, with Peter Gracar and Lukas Lüchtrath. Algorithms and Models for the Web Graph. 18th International Workshop, WAW 2023, Toronto, ON, Canada, May 23–26, 2023, Proceedings. Lecture Notes in Computer Science, vol 13894, 2023.

3. Transience Versus Recurrence for Scale-Free Spatial Networks, with Peter Gracar, Markus Heydenreich and Peter Mörters. Algorithms and Models for the Web Graph. 17th International Workshop, WAW 2020, Warsaw, Poland, September 21–22, 2020, Proceedings. Lecture Notes in Computer Science, vol 12091, 2020.

Working papers, extended abstracts and theses

1. Law of Large Numbers for an elementary model of Self-organised Criticality, with Antal A. Járai and Lorenzo Taggi. Working paper, 2023.

2. Conditionally Poissonian random digraphs. Working paper, 2017.

3. Persistence of activity in critical scale free Boolean networks. Tagungsbericht/ extended abstract, Oberwolfach Report 12 (2015), pp. 2020–2023.

4. Distances in preferential attachment networks. PhD thesis, December 2013, supervised by Prof. Peter Mörters, University of Bath.

5. Large deviations for the empirical pair measure of tree indexed Markov chains. Diploma thesis, April 2009, supervised by Prof. Heinrich von Weizsäcker, Technische Universität Kaiserslautern.