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Frank Redig

Publications and source records attributed to Frank Redig.

At least 19 recordsLinked to original sources
arXiv2026

Ergodic properties of the harmonic process

In this paper we study detailed fluctuation results for a class of non-equilibrium steady states. The main example is the boundary driven harmonic model \cite{frassek2022exact}. In this model, the non-equilibrium steady state (NESS) is a mixture of products of geometric distributions, of which the local parameters are in turn distributed as uniform order statistics. For such a NESS, we prove law of large numbers, central limit theorem and large deviation results for fields of a general local functions (generalizing the density field). We also obtain quantitative results on the deviation from local equilibrium.

math.PR↗
arXiv2026

Hydrodynamic limit of the symmetric exclusion process on complete Riemannian manifolds and principal bundles

We prove that the hydrodynamic limit of the symmetric exclusion process (SEP) is a Fokker-Planck equation in the setting of Poisson random neighborhood graphs approximating a weighted Riemannian manifold with Ricci curvature bounded from below. We also consider the lift of the SEP to a principal bundle, and obtain a Fokker-Planck equation with a weighted horizontal Laplacian as its hydrodynamic limit. Both results significantly extend the geometric settings in which one can prove the hydrodynamic limit from duality combined with convergence of the single particle random walk towards a diffusion process.

math.PR↗
arXiv2025

Non-equilibrium steady states with a spatial Markov structure

We investigate the structure of non-equilibrium steady states (NESS) for a class of exactly solvable models in the setting of a chain with left and right reservoirs. Inspired by recent results on the harmonic model, we focus on models in which the NESS is a mixture of equilibrium product measures, and where the probability measure which describes the mixture has a spatial Markovian property. We completely characterize the structure of such mixture measures, and show that under natural scaling and translation invariance properties, the only possible mixture measures are coinciding with the Dirichlet process found earlier in the context of the harmonic model.

math.PR↗
arXiv2025

Hydrodynamic limit and large deviations for run-and-tumble particles with mean-field switching rates

In this paper, we study run-and-tumble particles moving on two copies of the discrete torus (referred to as layers), where the switching rate between layers depends on a mean-field interaction among the particles. We derive the hydrodynamic limit of this model, as well as the large deviations from the hydrodynamic limit. Our main tool is the introduction of a weakly perturbed version of the system, whose hydrodynamic equations precisely characterize the trajectories associated with large deviations.

math.PR↗
arXiv2025

Random walks in a field of soft traps and criticality for the dissipative Abelian Sandpile Model

Motivated by the dissipative abelian sandpile model, we analyze the trajectories of a one-dimensional random walk in a landscape of soft traps. These traps, placed at increasing distances from each other, correspond to dissipative sites in the associated dissipative abelian sandpile model. We identify a critical growth rate of the sizes of intervals between successive traps where there is a transition between finiteness and non-finiteness of the expected survival time of the random walk. This corresponds to a transition between non-criticality and criticality of the associated dissipative abelian sandpile model. Therefore, in this setting, we thus identify precisely how much dissipation can be added to the original abelian sandpile model in order to disrupt its criticality.

math-ph↗
arXiv2025

Gaussian concentration bounds for probabilistic cellular automata

We study lattice spin systems and analyze the evolution of Gaussian concentration bounds (GCB) under the action of probabilistic cellular automata (PCA), which serve as discrete-time analogues of Markovian spin-flip dynamics. We establish the conservation of GCB and, in the high-noise regime, demonstrate that GCB holds for the unique stationary measure. Additionally, we prove the equivalence of GCB for the space-time measure and its spatial marginals in the case of contractive probabilistic cellular automata. Furthermore, we explore the relationship between (non)-uniqueness and GCB in the context of space-time Gibbs measures for PCA and illustrate these results with examples.

math.PR↗
arXiv2024

A large deviation principle for the multispecies stirring process

In this paper we consider the multispecies stirring process on the discrete torus. We prove a large deviation principle for the trajectory of the vector of densities of the different species. The technique of proof consists in extending the method of the foundational paper [1] based on the superexponential estimate to the multispecies setting. This requires a careful choice of the corresponding weakly asymmetric dynamics, which is parametrized by fields depending on the various species. We also prove the hydrodynamic limit of this weakly asymmetric dynamics, which is similar to but different from the ABC model in [2]. Using the appropriate asymmetric dynamics, we also obtain that the mobility matrix relating the drift currents to the fields coincides with the covariance matrix of the reversible multinomial distribution, which then further leads to the Einstein relation

math.PR↗
arXiv2024

Particle systems with sources and sinks

Local perturbations in conservative particle systems can have a non-local influence on the stationary measure. To capture this phenomenon, we analyze in this paper two toy models. We study the symmetric exclusion process on a countable set of sites V with a source at a given point (called the origin), starting from a Bernoulli product measure with density $ρ$. We prove that when the underlying random walk on V is recurrent, then the system evolves towards full occupation, whereas in the transient case we obtain a limiting distribution which is not product and has long-range correlations. For independent random walkers on V , we analyze the same problem, starting from a Poissonian measure. Via intertwining with a system of ODE's, we prove that the distribution is Poissonian at all later times t \> 0, and that the system ''explodes'' in the limit t $\rightarrow$ $\infty$ if and only if the underlying random walk is recurrent. In the transient case, the limiting density is a simple function of the Green's function of the random walk.

math.PR↗
arXiv2024

Intertwining and propagation of mixtures for generalized KMP models and harmonic models

We study a class of stochastic models of mass transport on discrete vertex set $V$. For these models, a one-parameter family of homogeneous product measures $\otimes_{i\in V} ν_θ$ is reversible. We prove that the set of mixtures of inhomogeneous product measures with equilibrium marginals, i.e., the set of measures of the form \[ \int\Big(\bigotimes_{i\in V} ν_{θ_i}\Big) \,Ξ(\prod_{i\in V}dθ_i) \] is left invariant by the dynamics in the course of time, and the ``mixing measure'' $Ξ$ evolves according to a Markov process which we then call ``the hidden parameter model''. This generalizes results from [7] to a larger class of models and on more general graphs. The class of models includes discrete and continuous generalized KMP models, as well as discrete and continuous harmonic models. The results imply that in all these models, the non-equilibrium steady state of their reservoir driven version is a mixture of product measures where the mixing measure is in turn the stationary state of the corresponding ``hidden parameter model''. For the boundary-driven harmonic models on the chain $\{1,\ldots, N\}$ with nearest neighbor edges, we recover that the stationary measure of the hidden parameter model is the joint distribution of the ordered Dirichlet distribution (cf. [3]), with a purely probabilistic proof based on a spatial Markov property of the hidden parameter model.

math.PR↗
arXiv2024

Stationary fluctuations of run-and-tumble particles

We study the stationary fluctuations of independent run-and-tumble particles. We prove that the joint densities of particles with given internal state converges to an infinite dimensional Ornstein-Uhlenbeck process. We also consider an interacting case, where the particles are subjected to exclusion. We then study the fluctuations of the total density, which is a non-Markovian Gaussian process, and obtain its covariance in closed form. By considering small noise limits of this non-Markovian Gaussian process, we obtain in a concrete example a large deviation rate function containing memory terms.

math.PR↗
arXiv2023

Large deviations and additivity principle for the open harmonic process

We consider the boundary driven harmonic model, i.e. the Markov process associated to the open integrable XXX chain with non-compact spins. Using the factorial moments we characterize the stationary measure as a mixture of product measures. For all spin values, we identify the law of the mixture in terms of the Dirichlet process. Next, by using the explicit knowledge of the non-equilibrium steady state we establish formulas predicted by Macroscopic Fluctuation Theory for several quantities of interest: the pressure (by Varadhan's lemma), the density large deviation function (by contraction principle), the additivity principle (by using the Markov property of the mixing law). To our knowledge, the results presented in this paper constitute the first rigorous derivation of these macroscopic properties for models of energy transport with unbounded state space, starting from the microscopic structure of the non-equilibrium steady state.

math.PR↗
arXiv2023

Invariance principle for Lifts of Geodesic Random Walks

We consider a certain class of Riemannian submersions $π: N \to M$ and study lifted geodesic random walks from the base manifold $M$ to the total manifold $N$. Under appropriate conditions on the distribution of the speed of the geodesic random walks, we prove an invariance principle; i.e., convergence to horizontal Brownian motion for the lifted walks. This gives us a natural probabilistic proof of the geometric identity relating the horizontal Laplacian $Δ_\H$ on $N$ and the Laplace-Beltrami operator $Δ_M$ on $M$. In particular, when $N$ is the orthonormal frame bundle $O(M)$, this identity is central in the Malliavin-Eells-Elworthy construction of Riemannian Brownian motion.

math.PR↗
arXiv2023

Density fluctuations for the multi-species stirring process

We study the density fluctuations at equilibrium of the multi-species stirring process, a natural multi-type generalization of the symmetric (partial) exclusion process. In the diffusive scaling limit, the resulting process is a system of infinite-dimensional Ornstein-Uhlenbeck processes that are coupled in the noise terms. This shows that at the level of equilibrium fluctuations the species start to interact, even though at the level of the hydrodynamic limit each species diffuses separately. We consider also a generalization to a multi-species stirring process with a linear reaction term arising from species mutation. The general techniques used in the proof are based on the Dynkin martingale approach, combined with duality for the computation of the covariances.

math.PR↗
arXiv2022

Concentration of Additive Functionals for Markov Processes and Applications to Interacting Particle Systems

We consider additive functionals of Markov processes in continuous time with general (metric) state spaces. We derive concentration bounds for their exponential moments and moments of finite order. Applications include diffusions, interacting particle systems and random walks. The method is based on coupling estimates and not spectral theory, hence reversibility is not needed. We bound the exponential moments(or the moments of finite order) in terms of a so-called coupled function difference, which in turn is estimated using the generalized coupling time. Along the way we prove a general relation between the contractivity of the semigroup and bounds on the generalized coupling time.

math.PR↗
arXiv2021

Boundary driven Markov gas: duality and scaling limits

Inspired by the recent work of Bertini and Posta, who introduced the boundary driven Brownian gas on $[0,1]$, we study boundary driven systems of independent particles in a general setting, including particles jumping on finite graphs and diffusion processes on bounded domains in $\mathbb{R}^d$. We prove duality with a dual process that is absorbed at the boundaries, thereby creating a general framework that unifies dualities for boundary driven systems in the discrete and continuum setting. We use duality first to show that from any initial condition the systems evolve to the unique invariant measure, which is a Poisson point process with intensity the solution of a Dirichlet problem. Second, we show how the boundary driven Brownian gas arises as the diffusive scaling limit of a system of independent random walks coupled to reservoirs with properly rescaled intensity.

math.PR↗
arXiv2021

Intertwining and Duality for Consistent Markov Processes

In this paper we derive intertwining relations for a broad class of conservative particle systems both in discrete and continuous setting. Using the language of point process theory, we are able to derive a natural framework in which duality and intertwining can be formulated. We prove falling factorial and orthogonal polynomial intertwining relations in a general setting. These intertwinings unite the previously found classical and orthogonal self-dualities in the context of discrete particle systems and provide new dualities for several interacting systems in the continuum. We also introduce a new process, the symmetric inclusion process in the continuum, for which our general method applies and yields generalized Meixner polynomials as orthogonal self-intertwiners.

math.PR↗
arXiv2021

Orthogonal polynomial duality of boundary driven particle systems and non-equilibrium correlations

We consider symmetric partial exclusion and inclusion processes in a general graph in contact with reservoirs, where we allow both for edge disorder and well-chosen site disorder. We extend the classical dualities to this context and then we derive new orthogonal polynomial dualities. From the classical dualities, we derive the uniqueness of the non-equilibrium steady state and obtain correlation inequalities. Starting from the orthogonal polynomial dualities, we show universal properties of n-point correlation functions in the non-equilibrium steady state for systems with at most two different reservoir parameters, such as a chain with reservoirs at left and right ends.

math.PR↗