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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Characterstic polynomials of random matrices and l
ogarithmically correlated processes - Fyodorov\, Y
(Queen Mary\, University of London)
DTSTART;TZID=Europe/London:20150423T100000
DTEND;TZID=Europe/London:20150423T110000
UID:TALK59152AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/59152
DESCRIPTION:I will discuss relations between logarithmically-c
orrelated Gaussian processes and the characteristi
c polynomials of large random $N imes N$ matric
es\, either from the Circular Unitary (CUE) or fro
m the Gaussian Unitary (GUE) ensembles. Such relat
ions help to address the problem of characterising
the distribution of the global maximum of the mod
ulus of such polynomials\, and of the Riemann $zet
aleft(rac{1}{2}+it\night)$ over some intervals of
$t$\n containing of the order of $log{t}$ zeroes
. I will show how to arrive to an explicit express
ion for the asymptotic probability density of the
maximum by combining the rigorous Fisher-Hartwig a
symptotics with the heuristic {it freezing transit
ion} scenario for logarithmically correlated proce
sses. Although the general idea behind the method
is the same for both CUE and GUE\, the latter case
is much more technically challenging. In particu
lar I will show how the conjectured {it self-dual
ity} in the freezing transition scenario plays the
crucial role in selecting the form of the maximum
distribution for GUE case. The found probability
densities will be compared to the results of dire
ct numerical simulations of the maxima. The presen
tation is mainly based on joint works with Ghaith
Hiary\, Jon Keating\, Boris Khoruzhenko\, and Nick
Simm.\n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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