{
  "id": "1007.4069",
  "version": "v2",
  "published": "2010-07-23T08:37:41.000Z",
  "updated": "2011-06-09T09:03:32.000Z",
  "title": "Self-intersection local times of random walks: Exponential moments in subcritical dimensions",
  "authors": [
    "Mathias Becker",
    "Wolfgang König"
  ],
  "comment": "15 pages. To appear in Probability Theory and Related Fields. The final publication is available at springerlink.com",
  "doi": "10.1007/s00440-011-0377-0",
  "categories": [
    "math.PR"
  ],
  "abstract": "Fix $p>1$, not necessarily integer, with $p(d-2)<d$. We study the $p$-fold self-intersection local time of a simple random walk on the lattice $\\Z^d$ up to time $t$. This is the $p$-norm of the vector of the walker's local times, $\\ell_t$. We derive precise logarithmic asymptotics of the expectation of $\\exp\\{\\theta_t \\|\\ell_t\\|_p\\}$ for scales $\\theta_t>0$ that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of $t$ and $\\theta_t$, and the precise rate is characterized in terms of a variational formula, which is in close connection to the {\\it Gagliardo-Nirenberg inequality}. As a corollary, we obtain a large-deviation principle for $\\|\\ell_t\\|_p/(t r_t)$ for deviation functions $r_t$ satisfying $t r_t\\gg\\E[\\|\\ell_t\\|_p]$. Informally, it turns out that the random walk homogeneously squeezes in a $t$-dependent box with diameter of order $\\ll t^{1/d}$ to produce the required amount of self-intersections. Our main tool is an upper bound for the joint density of the local times of the walk.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-06-09T09:03:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K37",
      "60F10",
      "60J55"
    ],
    "keywords": [
      "exponential moments",
      "subcritical dimensions",
      "fold self-intersection local time",
      "simple random walk",
      "walkers local times"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.4069B"
    }
  }
}