{
  "id": "2306.13936",
  "version": "v1",
  "published": "2023-06-24T10:54:49.000Z",
  "updated": "2023-06-24T10:54:49.000Z",
  "title": "Rate of convergence of the critical point of the memory-$τ$ self-avoiding walk in dimensions $d>4$",
  "authors": [
    "Noe Kawamoto"
  ],
  "comment": "26 pages, 4 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the spread-out models of the self-avoiding walk and its finite-memory version, called the memory-$\\tau$ walk. For both models, each step is uniformly distributed over the d-dimensional box $\\{x\\in\\mathbb Z^d:\\|x\\|_{\\infty} \\le L\\}$. The critical point $p_c^\\tau$ for the memory-$\\tau$ walk is increasing in $\\tau$ and converges to the critical point $p_c^\\infty$ for the self-avoiding walk as $\\tau\\uparrow\\infty$. The best estimate of the rate of convergence so far was obtained by Madras and Slade in [Lemma 6.8.6,The Self-Avoiding Walk,(Birkh\\\"auser, 1993)]: $p_c^{\\infty}-p_c^{\\tau} \\le K\\tau^{-(1+\\delta)}$, where $\\delta < (d-4)/2 \\wedge 1$ and $K<\\infty$ may depend on $d,~\\delta,~L$ but not on $\\tau$. By using the lace expansion, we show that there is a constant $C_d$ for $d>4$ and $L$ sufficiently large such that as $\\tau\\uparrow \\infty$, \\begin{equation} p_c^{\\infty}-p_c^{\\tau} =C_d\\tau^{-\\frac{d-2}{2}}+O(\\tau^{-\\frac{d-2}{2}}(\\log \\tau)^{-1}). \\end{equation} Moreover, we show that as $L \\uparrow \\infty$, $C_d=\\frac{2}{d-2}\\left(\\frac{d}{2\\pi\\Sigma_U^2}\\right)^{d/2}L^{-d}+O(L^{-d-1})$ where $\\Sigma_U^2$ is the variance of the uniform distribution over the $d$-dimensional box of side-length $2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-24T10:54:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical point",
      "self-avoiding walk",
      "convergence",
      "dimensions",
      "finite-memory version"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}