Welcome to my personal page! I am a mathematician and a former member of the Stochastics Group at the University of Mainz, where I was PI in the DFG-funded projects Inhomogeneous long-range percolation beyond the weak decay regime and Structure and dynamics of directed scale-free networks within Priority Programme 2265 Random Geometric Systems.
My scientific interests are at the intersection of probability, statistical physics, analysis and combinatorics. In winter 2022/23, I was acting professor (Chair of Stochastics and its Applications) at the University of Augsburg, and in 2019/20 I was a postdoc with Lisa Hartung. Before coming to Mainz, I worked several years in the group of Frank Aurzada at Technische Universität Darmstadt, and spent 3 terms lecturing Mathematical Finance at the University of Mannheim. My PhD research was conducted at the University of Bath under the supervision of Peter Mörters.
You can contact me at cmoench25+at+gmail.com.
scale-free networks
spatially extendend networks
directed random graphs
stochastic processes on (random) graphs
Inhomogenous long-range networks: connectivity & dynamics. This project is a continuation of my work within the DFG-Priority Programme SPP 22065 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 inhomogneity has on the fixation of the system.
This area of probability theory is concerned with problems of the following type: Given a real valued stochastic process whose range almost surely contains (a dense subset of) the whole real line, how rare is the event that it stays, say, below a given threshold for a certain period of time? For Markov processes the answer is given by classical fluctuation theory, but for non-Markovian processes the picture is very far from completion.
For an introduction / survey of this field see this article by F. Aurzada & T. Simon.
Strong order of the persistence probability of Fractional Brownian Motion. A well known result of G. Molchan states that the probability P(T) that standard fractional Brownian motion with Hurst index H does not hit a fixed level befor time T satisfies log(P(T))/log(T)~-(H-1) as T gets large. Our aim is to prove that there is also a power function f such that P(T)/f(T) converges.
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
Phase transitions for contact processes on sparse random graphs via metastability and local limits, with Benedikt Jahnel and Lukas Lüchtrath. arXiv:2505.22471
Loop vs. Bernoulli percolation on trees: strict inequality of critical values, with Andreas Klippel and Ben Lees. arXiv:2503.03319
Phase transitions for contact processes on one-dimensional networks, with Benedikt Jahnel and Lukas Lüchtrath. arXiv:2501.16858
Inhomogeneous long-range percolation in the strong decay regime: recurrence in one dimension. arXiv2408.06918
The mean field stubborn voter model, with Lisa Hartung and Florian Völlering. arXiv:2405.08202
The critical window in activated random walk on the complete graph, with Antal A. Járai and Lorenzo Taggi. arXiv2304.10169
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.
A very short proof of Sidorenko’s inequality for counts of homomorphism between graphs, with Lukas Lüchtrath, Bulletin of the Australian Mathematical Society published online 2025:1-5.
Inhomogeneous long-range percolation in the weak decay regime. Probability Theory and Related Fields 189, 3-4 (2024), pp. 1129–1160. MR4771112
Self-similar co-ascent processes and Palm calculus. Stochastic Processes and their Applications 174, (2024) paper no. 104378, 10 pp. MR4746578
DAG-type Distributed Ledgers via Young-age Preferential Attachment, with Amr Rizk. Stochastic Systems 13.3 (2023), pp. 377-397. MR4650338
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
Universality for persistence exponents of local times of self-similar processes with stationary increments. Journal of Theoretical Probability, 35 (2022), pp. 1842–1862. MR4488560
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
Distances and large deviations in the spatial preferential attachment model, with Christian Hirsch. Bernoulli 26.2 (2020), pp. 927--947. MR4058356
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
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
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
Typical distances in ultrasmall random networks, with Steffen Dereich and Peter Mörters. Advances in Applied Probability, 44.2 (2012), pp. 583–601. MR2977409
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.
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.
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.
Law of Large Numbers for an elementary model of Self-organised Criticality, with Antal A. Járai and Lorenzo Taggi. Working paper, 2023.
Conditionally Poissonian random digraphs. Working paper, 2017.
Persistence of activity in critical scale free Boolean networks. Tagungsbericht/ extended abstract, Oberwolfach Report 12 (2015), pp. 2020–2023.
Distances in preferential attachment networks. PhD thesis, December 2013, supervised by Prof. Peter Mörters, University of Bath.
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.