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Benedikt Jahnel

Publications and source records attributed to Benedikt Jahnel.

At least 37 records · Page 2Linked to original sources
arXiv2025

Phase transitions for contact processes on one-dimensional networks

We study the survival/extinction phase transition for contact processes with quenched disorder. The disorder is given by a locally finite random graph with vertices indexed by the integers that is assumed to be invariant under index shifts and augments the nearest-neighbour lattice by additional long-range edges. We provide sufficient conditions that imply the existence of a subcritical phase and therefore the non-triviality of the phase transition. Our results apply to instances of scale-free random geometric graphs with any integrable degree distribution. The present work complements previously developed techniques to establish the existence of a subcritical phase on Poisson--Gilbert graphs and Poisson--Delaunay triangulations (Ménard et al., Ann. Sci. Éc. Norm. Supér., 2016), on Galton--Watson trees (Bhamidi et al., Ann. Probab., 2021) and on locally tree-like random graphs (Nam et al., Trans. Am. Math. Soc., 2022), all of which require exponential decay of the degree distribution. Two applications of our approach are particularly noteworthy: Firstly, for Gilbert graphs derived from stationary point processes on $\mathbb{R}$ marked with i.i.d. random radii, our results are sharp. We show that there is a non-trivial phase transition if and only if the graph is locally finite. Secondly, for independent Bernoulli long-range percolation on $\mathbb{Z}$, with coupling constants $J_{x,y}\asymp |x-y|^{-δ}$, we verify a conjecture of Can (Electron. Commun. Probab., 2015) stating the non-triviality of the phase transition whenever $δ>2$. We believe that the results are indicative of the behaviour of contact processes on spatial random graphs also in dimensions $d > 1$ as long as the degree distribution of the underlying network has at least finite $d$-th moment. We support this by proving that no phase transition exists if the $d$-th moment is infinite.

math.PR↗
arXiv2025

Functional limit theorems for edge counts in dynamic random connection hypergraphs

We introduce a dynamic random hypergraph model constructed from a bipartite graph. In this model, both vertex sets of the bipartite graph are generated by marked Poisson point processes. Vertices of both vertex sets are equipped with marks representing their weight that influence their connection radii. Additionally, we also assign the vertices of the first vertex set a birth-death process with exponential lifetimes and the vertices of the second vertex set a time instant representing the occurrence of the corresponding vertices. Connections between vertices are established based on the marks and the birth-death processes, leading to a weighted dynamic hypergraph model featuring power-law degree distributions. We analyze the edge-count process in two distinct regimes. In the case of finite fourth moments, we establish a functional central limit theorem for the normalized edge count, showing convergence to a Gaussian AR(2)-type process as the observation window grows. In the challenging case of the heavy-tailed regime with infinite variance, we prove convergence to a novel stable process that is not Lévy and not even Markov.

math.PR↗
arXiv2025

Throughput in inhomogeneous planar drainage networks

We consider navigation schemes on planar diluted lattices and semi lattices with one discrete and one continuous component. More precisely, nodes that survive inhomogeneous Bernoulli site percolation, or are placed as inhomogeneous Poisson points on shifted copies of $\mathbb{Z}$, forward their individually generated traffic to their respective closest neighbors to the left in the next layer. The resulting drainage network is a tree and we study the amount of traffic that goes through an increasing window at the origin. Our main results show that, properly rescaled, the total traffic, jointly with the total length of the contributing tree part, converges to the area under a time-inhomogeneous Brownian motion until it hits zero. The hitting time corresponds to the limiting maximal path length.

math.PR↗
arXiv2025

On the long-time behaviour of reversible interacting particle systems in one and two dimensions

By refining Holley's free energy technique, we show that, under quite general assumptions on the dynamics, the attractor of a (possibly non-translation-invariant) interacting particle system in one or two spatial dimensions is contained in the set of Gibbs measures if the dynamics admits a reversible Gibbs measure. In particular, this implies that there can be no reversible interacting particle system that exhibits time-periodic behaviour and that every reversible interacting particle system is ergodic if and only if the reversible Gibbs measure is unique. In the special case of non-attractive stochastic Ising models this answers a question due to Liggett.

math.PR↗
arXiv2025

Phase transitions for the Widom--Rowlinson model in random environments

We establish non-uniqueness regimes for the infinite-volume two-colored Widom--Rowlinson model based on inhomogeneous Poisson point processes with locally finite intensity measures featuring percolation. As an application, we provide almost-sure phase-transition results for the Widom--Rowlinson model based on translation-invariant and ergodic Cox point processes with stabilizing and non-stabilizing directing measures.

math.PR↗
arXiv2024

The Planar Lattice Two-Neighbor Graph Percolates

The k-neighbor graph is a directed percolation model on the hypercubic lattice Z d in which each vertex independently picks exactly k of its 2d nearest neighbors at random, and we open directed edges towards those. We prove that the 2-neighbor graph percolates on Z 2 , i.e., that the origin is connected to infinity with positive probability. The proof rests on duality, an exploration algorithm, a comparison to i.i.d. bond percolation under constraints as well as enhancement arguments. As a byproduct, we show that i.i.d. bond percolation with forbidden local patterns has a strictly larger percolation threshold than 1/2. Additionally, our main result provides further evidence that, in low dimensions, less variability is beneficial for percolation.

math.PR↗
arXiv2024

Existence of subcritical percolation phases for generalised weight-dependent random connection models

We derive a sufficient condition for the existence of a subcritical percolation phase for a wide range of continuum percolation models where each vertex is embedded into Euclidean space according to an iid-marked stationary Poisson point process. In contrast to many established models, the probability of existence of an edge may not only depend on the distance and the weights of its end vertices but also on a surrounding vertex set. Our results can be applied in particular to models combining heavy-tailed degree distributions and long-range effects, which are typically well connected. More precisely, we study the critical annulus-crossing intensity $\widehatλ_c$ which is smaller or equal to the classical critical percolation intensity $λ_c$ and derive sharp conditions for $\widehatλ_c>0$ by controlling the occurrence of long edges. We further present tail bounds for the Euclidean diameter and number of points of the typical cluster in the subcritical phase and apply our results to several examples including some in which $\widehatλ_c<λ_c$.

math.PR↗
arXiv2024

Three-state $p$-SOS models on binary Cayley trees

We consider a version of the solid-on-solid model on the Cayley tree of order two in which vertices carry spins of value $0,1$ or $2$ and the pairwise interaction of neighboring vertices is given by their spin difference to the power $p>0$. We exhibit all translation-invariant splitting Gibbs measures (TISGMs) of the model and demonstrate the existence of up to seven such measures, depending on the parameters. We further establish general conditions for extremality and non-extremality of TISGMs in the set of all Gibbs measures and use them to examine selected TISGMs for a small and a large $p$. Notably, our analysis reveals that extremality properties are similar for large $p$ compared to the case $p=1$, a case that has been explored already in previous work. However, for the small $p$, certain measures that were consistently non-extremal for $p=1$ do exhibit transitions between extremality and non-extremality.

math-ph↗
arXiv2024

Percolation in lattice $k$-neighbor graphs

We define a random graph obtained via connecting each point of $\mathbb{Z}^d$ independently to a fixed number $1 \leq k \leq 2d$ of its nearest neighbors via a directed edge. We call this graph the directed $k$-neighbor graph. Two natural associated undirected graphs are the undirected and the bidirectional $k$-neighbor graph, where we connect two vertices by an undirected edge whenever there is a directed edge in the directed $k$-neighbor graph between them in at least one, respectively precisely two, directions. In these graphs we study the question of percolation, i.e., the existence of an infinite self-avoiding path. Using different kinds of proof techniques for different classes of cases, we show that for $k=1$ even the undirected $k$-neighbor graph never percolates, but the directed one percolates whenever $k \geq d+1$, $k \geq 3$ and $d \geq 5$, or $k \geq 4$ and $d=4$. We also show that the undirected $2$-neighbor graph percolates for $d=2$, the undirected $3$-neighbor graph percolates for $d=3$, and we provide some positive and negative percolation results regarding the bidirectional graph as well. A heuristic argument for high dimensions indicates that this class of models is a natural discrete analogue of the $k$-nearest-neighbor graphs studied in continuum percolation, and our results support this interpretation.

math.PR↗
arXiv2024

Gibbs measures for hardcore-SOS models on Cayley trees

We investigate the finite-state $p$-solid-on-solid model, for $p=\infty$, on Cayley trees of order $k\geq 2$ and establish a system of functional equations where each solution corresponds to a (splitting) Gibbs measure of the model. Our main result is that, for three states, $k=2,3$ and increasing coupling strength, the number of translation-invariant Gibbs measures behaves as $1\to3\to5\to6\to7$. This phase diagram is qualitatively similar to the one observed for three-state $p$-SOS models with $p>0$ and, in the case of $k=2$, we demonstrate that, on the level of the functional equations, the transition $p\to\infty$ is continuous.

math-ph↗
arXiv2023

Poisson approximation of fixed-degree nodes in weighted random connection models

We present a process-level Poisson-approximation result for the degree-k vertices in a high-density weighted random connection model with preferential-attachment kernel in the unit volume. Our main focus lies on the impact of the left tails of the weight distribution for which we establish general criteria based on their small-weight quantiles. To illustrate that our conditions are broadly applicable, we verify them for weight distributions with polynomial and stretched exponential left tails. The proofs rest on truncation arguments and a recently established quantitative Poisson approximation result for functionals of Poisson point processes.

math.PR↗
arXiv2023

A long-range contact process in a random environment

We study survival and extinction of a long-range infection process on a diluted one-dimensional lattice in discrete time. The infection can spread to distant vertices according to a Pareto distribution, however spreading is also prohibited at random times. We prove a phase transition in the recovery parameter via block arguments. This contributes to a line of research on directed percolation with long-range correlations in nonstabilizing random environments.

math.PR↗
arXiv2023

Continuum Percolation in a Nonstabilizing Environment

We prove phase transitions for continuum percolation in a Boolean model based on a Cox point process with nonstabilizing directing measure. The directing measure, which can be seen as a stationary random environment for the classical Poisson--Boolean model, is given by a planar rectangular Poisson line process. This Manhattan grid type construction features long-range dependencies in the environment, leading to absence of a sharp phase transition for the associated Cox--Boolean model. The phase transitions are established under individually as well as jointly varying parameters. Our proofs rest on discretization arguments and a comparison to percolation on randomly stretched lattices established in Hoffman 2005.

math.PR↗
arXiv2023

Trajectorial dissipation of $Φ$-entropies for interacting particle systems

A classical approach for the analysis of the longtime behavior of Markov processes is to consider suitable Lyapunov functionals like the variance or more generally $Φ$-entropies. Via purely analytic arguments it can be shown that these functionals are indeed non-increasing in time under quite general assumptions on the process. In this paper,we complement these classical results via a more probabilistic approach and show that dissipation is already present on the level of individual trajectories for spatially-extended systems of infinitely many interacting particles with arbitrary underlying geometry and compact local spin spaces. This extends previous results from the setting of finite-state Markov chains or diffusions in $\mathbb{R}^n$ to an infinite-dimensionalsetting with weak assumptions on the dynamics.

math.PR↗
arXiv2023

Limiting shape for First-Passage Percolation models on Random Geometric Graphs

Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variables on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape and we show that the shape is an Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. For the Richardson model we further show that it converges weakly to a nonstandard branching process in the joint limit of large intensities and slow passage times.

math.PR↗
arXiv2022

Chase-escape in dynamic device-to-device networks

The present paper features results on global survival and extinction of an infection in a multi-layer network of mobile agents. Expanding on a model first presented in [CHJW22], we consider an urban environment, represented by line-segments in the plane, in which agents move according to a random waypoint model based on a Poisson point process. Whenever two agents are at sufficiently close proximity for a sufficiently long time the infection can be transmitted and then propagates into the system according to the same rule starting from a typical device. Inspired by wireless network architectures, the network is additionally equipped with a second class of agents that is able to transmit a patch to neighboring infected agents that in turn can further distribute the patch, leading to a chase-escape dynamics. We give conditions for parameter configurations that guarantee existence and absence of global survival as well as an in-and-out of the survival regime, depending on the speed of the devices. We also provide complementary results for the setting in which the chase-escape dynamics is defined as an independent process on the connectivity graph. The proofs mainly rest on percolation arguments via discretization and multiscale analysis.

math.PR↗