## 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 language | English |
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Article number | 20120503 |

Pages (from-to) | 1 - 32 |

Number of pages | 32 |

Journal | Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 372 |

Issue number | 2007 |

Early online date | 16 Dec 2013 |

DOIs | |

Publication status | Published - 28 Jan 2014 |

## Keywords

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