{
  "id": "2501.13754",
  "version": "v1",
  "published": "2025-01-23T15:34:18.000Z",
  "updated": "2025-01-23T15:34:18.000Z",
  "title": "Large Deviations in Switching Diffusion: from Free Cumulants to Dynamical Transitions",
  "authors": [
    "Mathis Guéneau",
    "Satya N. Majumdar",
    "Gregory Schehr"
  ],
  "comment": "Letter: 6+2 pages and 2 figures; Supp. Mat.: 27 pages and 9 figures",
  "categories": [
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We study the diffusion of a particle with a time-dependent diffusion constant $D(t)$ that switches between random values drawn from a distribution $W(D)$ at a fixed rate $r$. Using a renewal approach, we compute exactly the moments of the position of the particle $\\langle x^{2n}(t) \\rangle$ at any finite time $t$, and for any $W(D)$ with finite moments $\\langle D^n \\rangle$. For $t \\gg 1$, we demonstrate that the cumulants $\\langle x^{2n}(t) \\rangle_c$ grow linearly with $t$ and are proportional to the free cumulants of a random variable distributed according to $W(D)$. For specific forms of $W(D)$, we compute the large deviations of the position of the particle, uncovering rich behaviors and dynamical transitions of the rate function $I(y=x/t)$. Our analytical predictions are validated numerically with high precision, achieving accuracy up to $10^{-2000}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-23T15:34:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "large deviations",
      "free cumulants",
      "dynamical transitions",
      "switching diffusion",
      "time-dependent diffusion constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 2,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}