{
  "id": "2202.04906",
  "version": "v1",
  "published": "2022-02-10T08:58:48.000Z",
  "updated": "2022-02-10T08:58:48.000Z",
  "title": "Number of distinct sites visited by a resetting random walker",
  "authors": [
    "Marco Biroli",
    "Francesco Mori",
    "Satya N. Majumdar"
  ],
  "comment": "41 pages, 8 figs",
  "categories": [
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We investigate the number $V_p(n)$ of distinct sites visited by an $n$-step resetting random walker on a $d$-dimensional hypercubic lattice with resetting probability $p$. In the case $p=0$, we recover the well-known result that the average number of distinct sites grows for large $n$ as $\\langle V_0(n)\\rangle\\sim n^{d/2}$ for $d<2$ and as $\\langle V_0(n)\\rangle\\sim n$ for $d>2$. For $p>0$, we show that $\\langle V_p(n)\\rangle$ grows extremely slowly as $\\sim \\left[\\log(n)\\right]^d$. We observe that the recurrence-transience transition at $d=2$ for standard random walks (without resetting) disappears in the presence of resetting. In the limit $p\\to 0$, we compute the exact crossover scaling function between the two regimes. In the one-dimensional case, we derive analytically the full distribution of $V_p(n)$ in the limit of large $n$. Moreover, for a one-dimensional random walker, we introduce a new observable, which we call imbalance, that measures how much the visited region is symmetric around the starting position. We analytically compute the full distribution of the imbalance both for $p=0$ and for $p>0$. Our theoretical results are verified by extensive numerical simulations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-10T08:58:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "full distribution",
      "one-dimensional random walker",
      "dimensional hypercubic lattice",
      "standard random walks",
      "exact crossover scaling function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}