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, , n\, we can describe an STS with n squares using two permutations σ, τ ∈ Sₙ, where σ encodes how the squares are glued horizontally and τ encodes how the squares are glued vertically. Hence, a previously considered natural model for STSs with n squares is Sₙ × Sₙ with the uniform distribution. We modify this model to obtain a new one: We fix α ∈ [0,1] and let K_μ be a conjugacy class of Sₙ with at most n^α cycles. Then K_μ × Sₙ with the uniform distribution is a model for STSs with restricted horizontal gluings. 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.