Abstract
Consider $N$ balls initially placed in $L$ bins. At each time step take a ball from each non-empty bin and \emph{randomly} reassign the balls into the bins.We call this finite Markov chain \emph{General Repeated Balls into Bins} process. It is a discrete time interacting particles system with parallel updates. Assuming a \emph{quantitative} chaotic condition on the reassignment rule we prove a \emph{quantitative} propagation of chaos for this model. We furthermore study some equilibrium properties of the limiting nonlinear process.
Highlights
In recent years interacting stochastic processes with parallel updates has received an increasing interest in the scientific literature and in the probabilistic one
Propagation of chaos gives the link between the microscopic and the macroscopic level, in particular it says that the system, in the large scale limit, behaves as the components
For the Repeated Balls-intoBins (RBB) process in [5] we proved the propagation of chaos in the weak limit sense and without any quantitative estimate
Summary
In recent years interacting stochastic processes with parallel updates has received an increasing interest in the scientific literature and in the probabilistic one. Parallel updating rules introduce new difficulties in the study of these processes, in particular in many cases the dynamics is not reversible and the invariant measure is unknown This means that the description of the system at equilibrium and the behaviour of a huge number of interacting components is not generally available. For the GRBB process the random reassignment has a general distribution The systems in this class are conservative interacting particles systems, in discrete time, with parallel updates. The quantitative chaotic condition on the reassignment rule is strong but natural as it can be seen as the distance between a canonical and the corresponding grand canonical measure This problem is known in literature as equivalence of ensembles in the thermodynamic limit (see for example [4]). In the last section we study the long time behavior of the nonlinear process
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