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SUMMARY:Characterstic polynomials of random matrices and logarithmically c
 orrelated processes - Fyodorov\, Y (Queen Mary\, University of London)
DTSTART:20150423T090000Z
DTEND:20150423T100000Z
UID:TALK59152@talks.cam.ac.uk
CONTACT:42080
DESCRIPTION:I will discuss relations between logarithmically-correlated Ga
 ussian processes and the characteristic polynomials of  large  random $N 	
 imes N$ matrices\, either from the Circular Unitary (CUE) or from the Gaus
 sian Unitary (GUE) ensembles. Such relations help to address the problem o
 f characterising the distribution of the global maximum of the modulus of 
 such polynomials\, and of the Riemann $zetaleft(rac{1}{2}+it\night)$ over
  some intervals of $t$\n  containing of the order of $log{t}$ zeroes. I wi
 ll show how to arrive to an explicit expression for the asymptotic probabi
 lity density of the maximum by combining the rigorous Fisher-Hartwig asymp
 totics with the heuristic {it freezing transition} scenario for logarithmi
 cally correlated processes. Although the general idea behind the method is
  the same for both CUE and GUE\, the latter case is much more technically 
 challenging.  In particular I will show how the conjectured  {it self-dual
 ity} in the freezing transition scenario plays the crucial role in selecti
 ng the form of the maximum distribution for GUE case.  The found probabili
 ty densities will be compared to the results of direct numerical simulatio
 ns of the maxima. The presentation is mainly based on joint works with Gha
 ith Hiary\, Jon Keating\, Boris Khoruzhenko\, and Nick Simm.\n
LOCATION:Seminar Room 1\, Newton Institute
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