Ergodicity and long-time behavior of the Random Batch Method for interacting particle systems

Recurrence and Ergodicity of Interacting Particle Systems

Many interacting particle systems with short range interactions are not ergodic, but converge weakly towards a mixture of their ergodic invariant measures. The question arises whether a.s. the process eventually stays close to one of these ergodic states, or if it changes between the attainable ergodic states infinitely often (“recurrence”). Under the assumption that there exists a convergence–...

Multiple Random Walks and Interacting Particle Systems

We study properties of multiple random walks on a graph under variousassumptions of interaction between the particles.To give precise results, we make our analysis for random regular graphs.The cover time of a random walk on a random r-regular graph was studiedin [6], where it was shown with high probability (whp), that for r ≥ 3 thecover time is asymptotic to θrn lnn, where...

A Multiscale Method for Interacting Particle Systems

Motivated by biological ion channels, we develop a new multiscale method for interacting particle systems coupling a particle and a continuum model. The method aims at the computation of particle interactions on the molecular scale and therefore incorporates particle correlations induced by short and long range interactions. To construct the multiscale method we use a stochastic particle model ...

Diffusive Behavior of Interacting Particle Systems

From Interacting Particle Systems to Random Matrices

In this contribution we consider stochastic growth models in the Kardar-Parisi-Zhang universality class in 1+1 dimension. We discuss the large time distribution and processes and their dependence on the class on initial condition. This means that the scaling exponents do not uniquely determine the large time surface statistics, but one has to further divide into subclasses. Some of the fluctuat...