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SUMMARY:Different Flavors of Asymptotics in Random Permutations and Their 
 Impact on Computing Finite-Size Effects - Folkmar Bornemann (Technische Un
 iversität München)
DTSTART:20221004T140000Z
DTEND:20221004T160000Z
UID:TALK182687@talks.cam.ac.uk
DESCRIPTION:The distribution of the length of longest increasing subsequen
 ces in random permutations/involutions of the symmetric group S_n is posit
 ioned in a rich web of knowledge connecting\, e.g.\, constructive combinat
 orics\, random matrix theory\, integrals over classical groups\, Toeplitz/
 Hankel determinants\, Riemann-Hilbert problems\, Painlev&eacute\;/Chazy eq
 uations\, and operator determinants. The known techniques for establishing
  a meaningful double-scaling limit near the mode of the length distributio
 n (in terms of the Tracy-Widom distributions for beta=1\, 2\, 4) use a Tau
 berian argument\, called de-Poissonization\, which does not render itself 
 to establish asymptotic expansions. Recently Forrester and Mays have start
 ed studying the structure of finite-size effects numerically and visualize
 d the coarse form of the first such term based on data from Monte-Carlo si
 mulations 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 a
 ccessible asymptotics that gives blazingly fast\, surprisingly robust and 
 accurate results &mdash\; outperforming combinatorial methods and the rand
 om 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 expecte
 d value and variance of the length distribution\, exhibiting several more 
 terms than previously known.
LOCATION:Seminar Room 2\, Newton Institute
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