{
  "id": "math/0510648",
  "version": "v1",
  "published": "2005-10-29T17:22:26.000Z",
  "updated": "2005-10-29T17:22:26.000Z",
  "title": "Balls-in-bins with feedback and Brownian Motion",
  "authors": [
    "Roberto Oliveira"
  ],
  "comment": "28 pages",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-10-29T17:22:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "60J10",
      "60J20"
    ],
    "keywords": [
      "brownian motion",
      "balls-in-bins process",
      "probability",
      "large initial number",
      "first bin starts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....10648O"
    }
  }
}