A model for horizontally restricted random square-tiled surfaces

A square-tiled surface (STS) is a (finite, possibly branched) cover of the standard square-torus with possible branching over exactly 1 point. Alternately, STSs can be viewed as finitely many axis-parallel squares with sides glued in parallel pairs. After a labelling of the squares by $\{1, \dots, n\}$, we can describe an STS with $n$ squares using two permutations $\sigma, \tau \in S_n$, where $\sigma$ encodes how the squares are glued horizontally and $\tau$ encodes how the squares are glued vertically. Hence, a previously considered natural model for STSs with $n$ squares is $S_n \times S_n$ with the uniform distribution. We modify this model to obtain a new one: We fix $\alpha \in [0,1]$ and let $\mathcal{K}_{\mu_n}$ be a conjugacy class of $S_n$ with at most $n^\alpha$ cycles. Then $\mathcal{K}_{\mu_n} \times S_n$ with the uniform distribution is a model for STSs with restricted horizontal gluings. Since horizontal cycles of the $\sigma$ permutation are related to the number of maximal horizontal cylinders, this new model serves as a random model for STSs with at most $n^\alpha$ maximal horizontal cylinders. We deduce the asymptotic (as $n$ grows) number of components, genus distribution, most likely stratum and set of holonomy vectors of saddle connections for random STSs in this new model.