Freezing transitions and extreme values: random matrix theory, zeta (1/2+it) and disordered landscapes

Yan V. Fyodorov*, Jonathan P. Keating

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)

58 Citations (Scopus)
236 Downloads (Pure)

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 x N random unitary (circular unitary ensemble) matrices U-N; i.e. the extreme value statistics of p(N)(theta) when N -> infinity. In addition, we argue that it leads to multi-fractal-like behaviour in the total length mu(N)(x) of the intervals in which vertical bar p(N)(theta)vertical bar > 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.

Original languageEnglish
Article number20120503
Pages (from-to)1 - 32
Number of pages32
JournalPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume372
Issue number2007
Early online date16 Dec 2013
DOIs
Publication statusPublished - 28 Jan 2014

Keywords

  • random matrix theory
  • Riemann zeta function
  • extreme values
  • SPIN-GLASSES
  • FIELD
  • MULTIFRACTALITY
  • MAXIMUM
  • MOMENTS
  • SYSTEMS

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