Enhanced large-genus cylinder-distribution conjecture

Establish the enhanced large-genus asymptotic law for horizontal-cylinder counts on every non-hyperelliptic connected component of odd quadratic-differential strata with a uniformly bounded number of poles: prove concentration of all singularities on one horizontal and one vertical foliation leaf and uniform mod-Poisson convergence of the cylinder-count distribution with parameter $\lambda_d=\log(d)/2$, limiting function $\sqrt{\pi}/\Gamma(t/2)$, and a radius of convergence uniformly bounded above one.

Background

The authors define random square-tiled surfaces by taking the uniform distribution on surfaces with at most a prescribed number of squares and use stable-graph contributions to describe cylinder-type frequencies.

They formulate an enhanced version of a conjecture from earlier work for strata with at most a fixed number of poles. The conjecture predicts both geometric concentration of singularities and a uniform analytic limit law for the number of horizontal cylinders as the genus, equivalently the dimension, tends to infinity.

References

The probability that all singularities of a random square-tiled surface in $Q(\tilde{})$ are located at the same leaf of the horizontal foliation and at the same leaf of the vertical foliation tends to 1 as $g\to\infty$. There exists a constant $R > 1$ such that the distribution of $K_{\tilde}$ converges mod-Poisson with parameter $\lambda_d=\log(d)/2$, limiting function $\frac{\sqrt\pi}{\Gamma(t/2)}$ and radius $R$ uniformly for all non-hyperelliptic components of $Q(\tilde{})$ of dimension $d$.

— Volumes of odd strata of quadratic differentials  (2502.13121 - Duryev et al., 18 Feb 2025) in Conjecture 8.1, Section 8.1, Distribution of cylinders