Extrema of Logarithmically Correlated Processes, Characteristic Polynomials, and the Riemann Zeta Function


Monday 9 May 2016 – Friday 13 May 2016 

Venue: NSQI Seminar Room, G05, Ground Floor, The Bristol Centre for Nanoscience and Quantum Information (NSQI), Tyndall Avenue, Bristol BS8 1FD

Sponsored by: Heilbronn Institute for Mathematical Research

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Background: In the last few years there has been a considerable growth of interest in various research areas concerning high and extreme values of logarithmically correlated random fields and processes. In part this has been caused by a confluence of ideas from disparate fields, and this workshop will bring together experts from these areas to focus on building new cross-discipline collaborations. One line of research originated in the probability community, and has its roots in works of Bramson on extrema of branching random walks. It culminated in a series of recent results of Zeitouni and collaborators (2012–15) on the maxima of Gaussian Free Fields and more general log-correlated Gaussian fields. Essentially, they found that the limiting law of the shifted maximum is generically given by Gumbel distribution with a random shift related to the so-called “derivative martingale”. Unfortunately statistical characterization of the latter is rather indirect. In parallel, in the Statistical Mechanics community related questions were addressed from the point of view of the so-called “freezing transition” paradigm. That line of research originated in a work by Carpentier & Le Doussal in 2001, and was essentially developed by Fyodorov & Bouchaud (2008), and Fyodorov, Le Doussal & Rosso (2009–2012). Explicit analytical expressions for the maximum distribution in a few representative logarithmically correlated models were conjectured, thus effectively bypassing the “derivative martingale” problem. However, many features of the freezing transition scenario presently remain conjectural, though recently some predictions were put on the firm mathematical ground by Arguin & Zindy and Subag & Zeitouni. Further interest in extrema of log-correlated processes was boosted by Fyodorov & Keating (2012–14) who pointed out the relevance of freezing scenario for understanding statistics of high values of characteristic polynomials of random matrices, and conjecturally of the Riemann zeta function along the critical line. The latter is a long-standing problem, see e.g. Farmer, Gonek, & Hughes (2008) where a conjecture on the extreme values of the zeta function was first developed, and references therein. Although the log-correlated nature of characteristic polynomials appeared already in the paper by Hughes, Keating & O’Connell (2001), it has only been used in the large-values context recently. Most of the picture developed by Fyodorov & Keating remains conjectural, and calls for a rigorous verification (or falsification) both for the random characteristic polynomials, and even more for the Riemann zeta function. From that angle, a very recent promising development comes from the work of Arguin, Belius and Harper (2015), which provides insights into the log-correlated nature of the Riemann zeta function, and associated extrema, in the framework of a nice toy model.

 

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