Propagation of Chaos for a Balls into Bins Model

TL;DR

This paper proves that in a balls-into-bins model with many bins, the distribution of balls in each bin becomes independent over time, demonstrating propagation of chaos, and explores equilibrium properties of the limiting process.

Contribution

It establishes propagation of chaos for a finite Markov chain model of balls into bins as the number of bins grows large, with analysis of equilibrium behavior.

Findings

Numbers of balls in each bin become independent as bins grow large

Propagation of chaos is proven for the model starting from chaotic initial states

Equilibrium properties of the limiting nonlinear process are analyzed

Abstract

Consider a finite number of balls initially placed in $L$ bins. At each time step a ball is taken from each non-empty bin. Then all the balls are uniformly reassigned into bins. This finite Markov chain is called Repeated Balls-into-Bins process and is a discrete time interacting particle system with parallel updating. We prove that, starting from a suitable (chaotic) set of initial states, as $L\to+\infty$, the numbers of balls in each bin becomes independent from the rest of the system i.e. we have propagation of chaos. We furthermore study some equilibrium properties of the limiting nonlinear process.