{
  "id": "2410.10754",
  "version": "v2",
  "published": "2024-10-14T17:26:27.000Z",
  "updated": "2024-10-15T07:40:23.000Z",
  "title": "The macroscopic shape of Gelfand-Tsetlin patterns and free probability",
  "authors": [
    "Samuel G. G. Johnston",
    "Joscha Prochno"
  ],
  "comment": "56 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "A Gelfand-Tsetlin function is a real-valued function $\\phi:C \\to \\mathbb{R}$ defined on a finite subset $C$ of the lattice $\\mathbb{Z}^2$ with the property that $\\phi(x) \\leq \\phi(y)$ for every edge $\\langle x,y \\rangle$ directed north or east between two elements of $C$. We study the statistical physics properties of random Gelfand-Tsetlin functions from the perspective of random surfaces, showing in particular that the surface tension of Gelfand-Tsetlin functions at gradient $u = (u_1,u_2) \\in \\mathbb{R}_{>0}^2$ is given by \\begin{align*} \\sigma(u_1,u_2) = - \\log (u_1 + u_2 ) - \\log \\sin (\\pi u_1/(u_1+u_2)) -1 + \\log \\pi. \\end{align*} A Gelfand-Tsetlin pattern is a Gelfand-Tsetlin function defined on the triangle $T_n = \\{(x_1,x_2) \\in \\mathbb{Z}^2 : 1 \\leq x_2 \\leq x_1 \\leq n \\}$. We show that after rescaling, a sequence of random Gelfand-Tsetlin patterns with fixed diagonal heights approximating a probability measure $\\mu$ satisfies a large deviation principle with speed $n^2$ and rate functional of the form \\begin{align*} \\mathcal{E}[\\psi] := \\int_{\\blacktriangle} \\sigma(\\nabla \\psi)\\, \\mathrm{d}s \\,\\mathrm{d}t - \\chi[\\mu] \\end{align*} where $\\chi[\\mu]$ is Voiculescu's free entropy. We show that the Euler-Lagrange equations satisfied by the minimiser of the rate functional agree with those governing the free compression operation in free probability, thereby resolving a recent conjecture of Shlyakhtenko and Tao.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-10-15T07:40:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B41",
      "82B20",
      "46L54",
      "15A42",
      "60G55",
      "60F10",
      "49Q20"
    ],
    "keywords": [
      "free probability",
      "macroscopic shape",
      "random gelfand-tsetlin functions",
      "large deviation principle",
      "rate functional agree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 56,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}