Near-Involutions, the Pillowcase Distribution, and Quadratic Differentials

TL;DR

This paper explores the role of near-involutions and pillowcase weights in calculating volumes of quadratic differential moduli spaces, revealing distribution properties and limit shapes of Young diagrams.

Contribution

It provides explicit formulas for near-involutions and analyzes the pillowcase distribution, connecting these to the geometry of quadratic differentials.

Findings

Limit shape matches the uniform distribution

Probability concentrates on partitions with similar 2-quotients

No full Central Limit Theorem applies

Abstract

In the context of A. Eskin and A. Okounkov's approach to the calculation of the volumes of the different strata of the moduli space of quadratic differentials, two objects have a prominent role. Namely, the characters of near-involutions and the pillowcase weights. For the former we give a fairly explicit formula. On the other hand, the pillowcase weights induce a distribution on the space of Young diagrams. We analyze this distribution and prove several facts, including that its limit shape corresponds to the one induced by the uniform distribution, that the probability concentrates on the set of partitions with very similar 2-quotients, and that there is no hope for a full Central Limit Theorem. This is a reformatted version of the author's Ph.D. thesis, advised by Professor Andrei Okounkov. The results will be published in a forthcoming paper.