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UID:6702@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20200710T110000
DTEND;TZID=Europe/Paris:20200710T120000
DTSTAMP:20241120T201942Z
URL:https://www.i2m.univ-amu.fr/evenements/geometrie-asymptotique-de-surfa
 ces-a-petits-carreaux-aleatoires-et-de-multi-courbes-aleatoires-en-grand-g
 enre/
SUMMARY:Anton Zorich (IMJ-PRG\, Paris): Géométrie asymptotique de surface
 s à petits carreaux aléatoires et de multi-courbes aléatoires en grand 
 genre
DESCRIPTION:Anton Zorich: Anton ZORICH\nWe study the combinatorial geometry
  of a random closed multicurve on a surface of large genus $g$ and of a ra
 ndom square-tiled surface of large genus $g$. We prove that primitive comp
 onents $\\gamma_1\, \\dots\,\\gamma_k$ of a random multicurve $m_1\\gamma_
 1+\\dots +m_k\\gamma_k$ represent linearly independent homology cycles wit
 h asymptotic probability $1$ and that all its weights $m_i$ are equal to $
 1$ with asymptotic probability $\\sqrt{2}/2$. We prove analogous propertie
 s for random square-tiled surfaces. In particular\, we show that all conic
 al singularities of a random square-tiled surface belong to the same leaf 
 of the horizontal foliation and to the same leaf of the vertical foliation
  with asymptotic probability $1$.\nWe show that the number of components o
 f a random multicurve and the number of maximal horizontal cylinders of a 
 random square-tiled surface of genus $g$ are both very well-approximated b
 y the number of cycles of a random permutation for an explicit non-uniform
  measure on the symmetric group of $3g-3$ elements. In particular\, we pro
 ve that the expected value of these quantities is asymptotically equivalen
 t to $(\\log(6g-6) + \\gamma)/2 + \\log 2$.\nThese results are based on ou
 r formula for the Masur--Veech volume $\\Vol\\cQ_g$ of the moduli space of
  holomorphic quadratic differentials combined with deep large genus asympt
 otic analysis of this formula performed by A. Aggarwal and with the unifor
 m asymptotic formula for intersection numbers of $\\psi$-classes on $\\ove
 rline{\\cM}_{g\,n}$ for large $g$ proved by A.~Aggarwal in 2020.
ATTACH;FMTTYPE=image/jpeg:https://www.i2m.univ-amu.fr/wp-content/uploads/2
 020/06/Anton_Zorich.jpg
CATEGORIES:Séminaire,Rauzy
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