Ricerc@Sapienza

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 → +∞, the numbers of balls in each bin become independent from the rest of the system i.e. we have propagation of chaos.

We consider a nonreversible finite Markov chain called Repeated Balls-into-Bins (RBB) process. This process is a discrete time conservative interacting particle system with parallel updates. Place initially in L bins rL balls, where r is a fixed positive constant. At each time step a ball is removed from each non-empty bin. Then all these removed balls are uniformly reassigned into bins. We prove that the mixing time of the RBB process is of order L. Furthermore we show that if the initial configuration has o(L) balls per site the equilibrium is attained in o(L) steps.