Balls-in-bins with feedback and Brownian Motion
In a balls-in-bins process with feedback, balls are sequentially thrown into bins so that the probability that a bin with n balls obtains the next ball is proportional to f(n) for some function f. A commonly studied case where there are two bins and f(n) = n^p for p > 0, and our goal is to study the fine behavior of this process with two bins and a large initial number t of balls. Perhaps surprisingly, Brownian Motions are an essential part of both our proofs. For p>1/2, it was known that with probability 1 one of the bins will lead the process at all large enough times. We show that if the first bin starts with t+\lambda\sqrt{t} balls (for constant \lambda\in \R), the probability that it always or eventually leads has a non-trivial limit depending on \lambda. For p\leq 1/2, it was known that with probability 1 the bins will alternate in leadership. We show, however, that if the initial fraction of balls in one of the bins is >1/2, the time until it is overtaken by the remaining bin scales like \Theta({t^{1+1/(1-2p)}}) for p<1/2 and \exp(\Theta{t}) for p=1/2. In fact, the overtaking time has a non-trivial distribution around the scaling factors, which we determine explicitly. Our proofs use a continuous-time embedding of the balls-in-bins process (due to Rubin) and a non-standard approximation of the process by Brownian Motion. The techniques presented also extend to more general functions f.