arXiv subjects

Jonas Köppl

Publications and source records attributed to Jonas Köppl.

At least 19 recordsLinked to original sources

arXiv2026

Comparison inequalities for discrete- and continuous-time infection processes

We study comparison inequalities for contact-process-type infection dynamics on $\mathbb Z^d$ with general local transmission mechanisms. The processes considered include both discrete-time oriented-percolation models and continuous-time contact processes in which infection events may transmit to random, possibly correlated sets of neighbouring sites. Our main results apply local-to-global comparison criteria beyond stochastic domination: if one local infection rule is more likely than another to hit every non-empty test set of neighbours, then the corresponding process has larger survival probabilities at all times. We compare classical independent-infection models with exchangeable fixed-budget, all-or-nothing, and burst-type infection mechanisms, deriving orderings for example of finite-time and infinite-time survival probabilities and stopping-time distributions. In continuous time, we further obtain consequences for critical survival thresholds of exchangeable infection laws. The results show that infection processes with identical or comparable mean-field infection intensity can nevertheless be rigorously ordered at the stochastic level once spatial geometry is taken into account. Additionally, it turns out that the classical continuous-time contact process dominates a variety of related continuous-time models, whereas its most natural discrete-time version, Bernoulli oriented percolation, is not dominant with respect to related discrete-time models.

math.PR↗

arXiv2026

On the stationary measures of the critical Ornstein--Uhlenbeck process

We study linear and symmetric diffusion processes on $\mathbb{R}^{\mathbb{Z}^d}$ that can be seen as the Langevin dynamics of the (inhomogeneous) harmonic crystal for given conductances, with a particular focus on their stationary distributions. Despite the linearity of the interaction, we exhibit a rich variety of behaviours, depending on the disorder encoded by the conductances. Our main results provide essentially sharp criteria on the conductances ensuring that every stationary measure is reversible. We also provide examples for which these criteria fail and where either no stationary measures exist or reversible and non-reversible stationary measures coexist. Additionally, we show that the linear system can even exhibit non-trivial time-periodic behaviour and provide a spectral characterisation of the occurrence of such oscillations. The case of deterministic conductances is complemented by a study of the case of random conductances under quite general moment assumptions.

math.PR↗

arXiv2026

Asymptotically complete free-energy dissipation: a coarse MLSI holds at any positive temperature

Everybody learns in school that an out-of-equilibrium system coupled to a heat bath at a fixed temperature evolves to thermodynamic equilibrium as time goes on, and the free energy will only decrease on its way there. At least since Holley's 1971 work, mathematicians know this too, in the modest context of classical Ising Glauber dynamics. But does the free energy also asymptotically decrease to the free energy of an equilibrium state? A typical school kid would say "yes, of course", but since the free energy is only lower semicontinuous, this question is less straightforward than it initially seems. To the best of our knowledge, apart from the uniqueness regime, where one can make use of classical functional inequalities, this question has not previously been resolved rigorously. We prove the asymptotically complete dissipation of the free energy for the classical Ising Glauber dynamics by introducing a coarse modified log-Sobolev inequality, which holds at every positive temperature, in particular in the phase-coexistence regime.

math.PR↗

arXiv2026

Convex order and faster transmission in first contact percolation

Inspired by strict-monotonicity criteria for the time constant in first passage percolation, we investigate convex ordering of point processes in relation to the time constant in first contact percolation. In a nutshell, first contact percolation models the spread of an infection as a contact process without recovery based on a generalized graphical representation, where the usual homogeneous Poisson point processes on the edges are replaced by general simple point processes. Based on a notion of convex ordering for point processes, we prove monotonicity in the number and existence of infection paths. We argue that this convex ordering is however not enough to ensure strict monotonicities in the asymptotic speed of the infection. Instead, we propose a criterion based on an ordering of void probabilities and prove a speed-up for one-dimensional systems based on $\mathbb{Z}$-stationary point processes.

math.PR↗

arXiv2026

Restriction and mixing properties of interacting particle systems with unbounded range

We consider interacting particle systems with unbounded interaction range on general countably infinite graphs $S$ and prove explicit non-asymptotic error bounds for approximations of the infinite-volume dynamics by systems of finitely many interacting particles. Moreover, we also provide non-asymptotic quantitative bounds on the spatial decay of correlations at times $t>0$ and then apply these results to show that interacting particle systems on $\mathbb{Z}$ whose interaction strengths decays exponentially fast cannot spontaneously break the time-translation symmetry, neither in the strong, nor in the weak sense.

math.PR↗

arXiv2026

Local criteria for global connectivity comparisons: beyond stochastic domination

We introduce a site-wise domination criterion for local percolation models, which enables the comparison of one-arm probabilities even in the absence of stochastic domination. The method relies on a local-to-global principle: if, at each site, one model is more likely than the other to connect to a subset of its neighbors, for all nontrivial such subsets, then this advantage propagates to connectivity events at all scales. In this way, we obtain a robust alternative to stochastic domination, applicable in all cases where the latter works and in many where it does not. As a main application, we compare classical Bernoulli bond percolation with degree-constrained models, showing that degree constraints enhance percolation, and obtain asymptotically optimal bounds on critical parameters for degree-constrained models.

math.PR↗

arXiv2026

On the absence of time-translation symmetry breaking in some non-reversible interacting particle systems

The conditions under which stochastic systems of infinitely many interacting particles can maintain sufficient spatial order to move coherently along a time-periodic orbit, thereby breaking the time-translation invariance of the underlying dynamical equation, have been an elusive issue. Via a free energy technique, we prove that if a non-reversible interacting particle system on $\mathbb{Z}^d$, $d=1,2$, with strictly positive rates admits a product measure as a stationary measure, then it cannot exhibit time-periodic behaviour. This provides a first step towards a general conjecture that time-periodic behaviour cannot occur in one- and two-dimensional systems with short-range interactions and constitutes the first result for non-reversible dynamics in dimension two.

math.PR↗

arXiv2026

Reversible birth-and-death dynamics in continuum: a de Bruijn-type identity for free-energy dissipation

We investigate free-energy dissipation in a continuous-time birth-and-death dynamics in $\mathbb{R}^d$. For these Markov processes, the class of reversible measures coincides with the infinite-volume Gibbs point processes for some sufficiently nice Hamiltonian. For a wide class of initial distributions, we derive a de~Bruijn-type identity that relates the time evolution of the specific relative entropy along trajectories to the Fisher information, in particular establishing the thermodynamic limit of the latter. Along the way, we analyze some fine properties of the considered dynamics, such as the existence and regularity of local densities, obtain a spatial ergodic theorem for the entropy production per unit volume, and derive a small-time exponential series expansion of the dynamics.

math.PR↗

arXiv2026

A correlation bound for the one-dimensional heterogeneous Ising model

We derive a new upper bound for the correlations in a heterogeneous one-dimensional Ising model with free boundary conditions. The new upper bound quantifies the simultaneous decay of correlations due to weakness of nearest-neighbor coupling constants and the effect of external fields. The proof constitutes an application of random currents to a non-ferromagnetic Ising model.

math.PR↗

arXiv2025

The variational principle for a marked Gibbs point process with infinite-range multibody interactions

We prove the Gibbs variational principle for the Asakura--Oosawa model in which particles of random size obey a hardcore constraint of non-overlap and are additionally subject to a temperature-dependent area interaction. The particle size is unbounded, leading to infinite-range interactions, and the potential cannot be written as a $k$-body interaction for fixed $k$. As a byproduct, we also prove the existence of infinite-volume Gibbs point processes satisfying the DLR equations. The essential control over the influence of boundary conditions can be established using the geometry of the model and the hard-core constraint.

math.PR↗

arXiv2025

Dynamical Gibbs Variational Principles for Irreversible Interacting Particle Systems with Applications to Attractor Properties

We consider irreversible translation-invariant interacting particle systems on the $d$-dimensional cubic lattice with finite local state space, which admit at least one Gibbs measure as a time-stationary measure. Under some mild degeneracy conditions on the rates and the specification we prove, that zero relative entropy loss of a translation-invariant measure implies, that the measure is Gibbs w.r.t. the same specification as the time-stationary Gibbs measure. As an application, we obtain the attractor property for irreversible interacting particle systems, which says that any weak limit point of any trajectory of translation-invariant measures is a Gibbs measure w.r.t. the same specification as the time-stationary measure. This extends previously known results to fairly general irreversible interacting particle systems.

math.PR↗

arXiv2025

Time-periodic behaviour in one- and two-dimensional interacting particle systems

We provide a class of examples of interacting particle systems on $\mathbb{Z}^d$, for $d\in\{1,2\}$, that admit a unique translation-invariant stationary measure, which is not the long-time limit of all translation-invariant starting measures, due to the existence of time-periodic orbits in the associated measure-valued dynamics. This is the first such example and shows that even in low dimensions, not every limit point of the measure-valued dynamics needs to be a time-stationary measure.

math.PR↗

arXiv2025

Reversible birth-and-death dynamics in continuum: free-energy dissipation and attractor properties

We consider continuous-time birth-and-death dynamics in $\mathbb{R}^d$ that admit at least one infinite-volume Gibbs point process based on area interactions as a reversible measure. For a large class of starting measures, we show that the specific relative entropy decays along trajectories, and that all possible long-time weak limit points are also Gibbs point processes with respect to the same interaction. Our proof rests on a representation of the entropy dissipation in terms of the Palm version of the propagated measure.

math.PR↗

arXiv2025

Survival and extinction for a contact process with a density-dependent birth rate

To study later spatial evolutionary games based on the multitype contact process, we first focus in this paper on the conditions for survival/extinction in the presence of only one strategy, in which case our model consists of a variant of the contact process with a density-dependent birth rate. The players are located on the $d$-dimensional integer lattice, with natural birth rate $λ$ and natural death rate one. The process also depends on a payoff $a_{11} = a$ modeling the effects of the players on each other: while players always die at rate one, the rate at which they give birth is given by $λ$ times the exponential of $a$ times the fraction of occupied sites in their neighborhood. In particular, the birth rate increases with the local density when $a > 0$, in which case the payoff $a$ models mutual cooperation, whereas the birth rate decreases with the local density when $a < 0$, in which case the payoff $a$ models intraspecific competition. Using standard coupling arguments to compare the process with the basic contact process (the particular case $a = 0$), we prove that, for all payoffs $a$, there is a phase transition from extinction to survival in the direction of $λ$. Using various block constructions, we also prove that, for all birth rates $λ$, there is a phase transition in the direction of $a$. This last result is in sharp contrast with the behavior of the nonspatial deterministic mean-field model in which the stability of the extinction state only depends on $λ$. This underlines the importance of space (local interactions) and stochasticity in our model.

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↗

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

Evolutionary games on the lattice: multitype contact process with density-dependent birth rates

Interacting particle systems of interest in evolutionary game theory introduced in the probability literature consist of variants of the voter model in which each site is occupied by one player. The goal of this paper is to initiate the study of evolutionary games based more realistically on the multitype contact process in which each site is either empty or occupied by a player following one of two possible competing strategies. Like in the symmetric multitype contact process, players have natural death rate one and natural birth rate $λ$. Following the traditional modeling approach of evolutionary game theory, the process also depends on a payoff matrix $A = (a_{ij})$ where $a_{ij}$ represents the payoff a type $i$ player receives from each of its type $j$ neighbors, and the actual birth rate is an increasing function of the payoff. Using various couplings and block constructions, we first prove the existence of a phase transition in the direction of the intra payoff $a_{11}$ or $a_{22}$ while the other three payoffs are fixed. We also look at the behavior near the critical point where all four payoffs are equal to zero, in which case the system reduces to the symmetric multitype contact process. The effects of the intra payoffs $a_{11}$ and $a_{22}$ are studied using various couplings and duality techniques, while the effects of the inter payoffs $a_{12}$ and $a_{21}$ are studied in one dimension using a coupling with the contact process to control the interface between the 1s and the 2s.

math.PR↗

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↗