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SUMMARY:Numerical Generatingfunctionology: Counting with Toeplitz Determin
 ants\, Hayman-Admissibility\, and the Wiener-Hopf-Factorization - Folkmar 
 Bornemann (Technische Universität München)
DTSTART:20191211T113000Z
DTEND:20191211T123000Z
UID:TALK135577@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:Counting related to representation theory and symmetric functi
 ons can be framed as generating functions given by Toeplitz determinants. 
 Prime examples are counting all permutations with no long increasing subse
 quence or lattice paths in last passage percolation. Intricate scaling lim
 its of those generating functions have been used\, e.g.\, in the seminal w
 ork by Baik/Deift/Johansson\, to obtain asymptotic formulae in terms of ra
 ndom matrix theory. In this talk\, we address the question whether generat
 ing functions can be used to numerically extract the counts in a mesoscopi
 c regime where combinatorial methods are already infeasible and the random
  matrix asymptotics is still too inaccurate. The stable computation of the
  counts by means of complex analysis is possible\, indeed\, and can be exp
 lained by the theory of Hayman admissibility. As a bonus track from comple
 x analysis\, the numerical evaluation of the Toeplitz determinant itself h
 as to be stabilized by a variant of the Borodin-Okounkov formula based on 
 the Wiener-Hopf factorization. This way\, we obtain\, e.g.\, exact 1135-di
 git counts in permutations of order 500 or\, by taking Hayman&rsquo\;s fam
 ous generalization of Stirling&rsquo\;s formula at face value\, a blazingl
 y fast\, surprisingly robust and accurate numerical asymptotics.
LOCATION:Seminar Room 1\, Newton Institute
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