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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Different Flavors of Asymptotics in Random Permuta
tions and Their Impact on Computing Finite-Size Ef
fects - Folkmar Bornemann (Technische Universität
München)
DTSTART;TZID=Europe/London:20221004T150000
DTEND;TZID=Europe/London:20221004T170000
UID:TALK182687AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/182687
DESCRIPTION:The distribution of the length of longest increasi
ng subsequences in random permutations/involutions
of the symmetric group S_n is positioned in a ric
h web of knowledge connecting\, e.g.\, constructiv
e combinatorics\, random matrix theory\, integrals
over classical groups\, Toeplitz/Hankel determina
nts\, Riemann-Hilbert problems\, Painlevé\;/
Chazy equations\, and operator determinants. The k
nown techniques for establishing a meaningful doub
le-scaling limit near the mode of the length distr
ibution (in terms of the Tracy-Widom distributions
for beta=1\, 2\, 4) use a Tauberian argument\, ca
lled de-Poissonization\, which does not render its
elf to establish asymptotic expansions. Recently F
orrester and Mays have started studying the struct
ure of finite-size effects numerically and visuali
zed the coarse form of the first such term based o
n data from Monte-Carlo simulations for n up to 10
^5. In this talk we show that the theory of Hayman
admissibility yields a different\, less explicit
but numerically highly accessible asymptotics that
gives blazingly fast\, surprisingly robust and ac
curate results &mdash\; outperforming combinatoria
l methods and the random matrix asymptotics in the
mesoscopic regime (for\, say\, n up to 10^{12}).
It allows to approximate the first two finite-size
corrections to the random matrix limit. We derive
\, heuristically\, expansions of the expected valu
e and variance of the length distribution\, exhibi
ting several more terms than previously known.
LOCATION:Seminar Room 2\, Newton Institute
CONTACT:
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