{
  "id": "1211.6063",
  "version": "v2",
  "published": "2012-11-26T18:54:23.000Z",
  "updated": "2013-12-23T22:42:05.000Z",
  "title": "Freezing Transitions and Extreme Values: Random Matrix Theory, $ζ(1/2+it)$, and Disordered Landscapes",
  "authors": [
    "Yan V. Fyodorov",
    "Jonathan P. Keating"
  ],
  "comment": "The published version",
  "journal": "Phil. Trans. R. Soc. A v.372 (2014), 20120503 [32 pages]",
  "doi": "10.1098/rsta.2012.0503",
  "categories": [
    "math-ph",
    "cond-mat.dis-nn",
    "math.MP",
    "math.NT",
    "math.PR"
  ],
  "abstract": "We argue that the freezing transition scenario, previously conjectured to occur in the statistical mechanics of 1/f-noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials p_N(\\theta) of large N\\times N random unitary (CUE) matrices; i.e. the extreme value statistics of p_N(\\theta) when N \\rightarrow\\infty. In addition, we argue that it leads to multifractal-like behaviour in the total length \\mu_N(x) of the intervals in which |p_N(\\theta)|>N^x, x>0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta-function \\zeta(s) over stretches of the critical line s=1/2+it of given constant length, and present the results of numerical computations of the large values of \\zeta(1/2+it). Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-12-23T22:42:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random matrix theory",
      "freezing transition",
      "disordered landscapes",
      "large values taken",
      "extreme value statistics"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Philosophical Transactions of the Royal Society of London Series A",
      "year": 2013,
      "month": "Dec",
      "volume": 372,
      "number": 2007,
      "pages": 20120503
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013RSPTA.37220503F"
    }
  }
}