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Quasimodularity and large genus limits of Siegel-Veech constants
Quasimodular forms were first studied in the context of counting torus
coverings. Here we show that a weighted version of these coverings with
Siegel-Veech weights also provides quasimodular forms. We apply this to prove
conjectures of Eskin and Zorich on the large genus limits of Masur-Veech
volumes and of Siegel-Veech constants.
In Part I we connect the geometric definition of Siegel-Veech constants both
with a combinatorial counting problem and with intersection numbers on Hurwitz
spaces. We introduce modified Siegel-Veech weights whose generating functions
will later be shown to be quasimodular.
Parts II and III are devoted to the study of the quasimodularity of the
generating functions arising from weighted counting of torus coverings. The
starting point is the theorem of Bloch and Okounkov saying that q-brackets of
shifted symmetric functions are quasimodular forms. In Part II we give an
expression for their growth polynomials in terms of Gaussian integrals and use
this to obtain a closed formula for the generating series of cumulants that is
the basis for studying large genus asymptotics. In Part III we show that the
even hook-length moments of partitions are shifted symmetric polynomials and
prove a formula for the q-bracket of the product of such a hook-length moment
with an arbitrary shifted symmetric polynomial. This formula proves
quasimodularity also for the (-2)-nd hook-length moments by extrapolation, and
implies the quasimodularity of the Siegel-Veech weighted counting functions.
Finally, in Part IV these results are used to give explicit generating
functions for the volumes and Siegel-Veech constants in the case of the
principal stratum of abelian differentials. To apply these exact formulas to
the Eskin-Zorich conjectures we provide a general framework for computing the
asymptotics of rapidly divergent power series.Comment: 107 pages, final version, to appear in J. of the AM
Enumeration of Standard Young Tableaux
A survey paper, to appear as a chapter in a forthcoming Handbook on
Enumeration.Comment: 65 pages, small correction
The pillowcase distribution and near-involutions
In the context of the Eskin-Okounkov approach to the calculation of the
volumes of the different strata of the moduli space of quadratic differentials,
the important ingredients are the pillowcase weight probability distribution on
the space of Young diagrams, and the asymptotic study of characters of
permutations that near-involutions. In this paper we present various new
results for these objects. Our results give light to unforeseen difficulties in
the general solution to the problem, and they simplify some of the previous
proofs.Comment: This paper elaborates on some of the results of the author's PhD
thesis (arXiv:1209.4333). This is the published version,
http://ejp.ejpecp.org/article/view/362
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