The Cheeger constants of random Belyi surfaces

Brooks and Makover developed a combinatorial model of random hyperbolic surfaces by gluing certain hyperbolic ideal triangles. In this paper we show that for any $ϵ>0$, as the number of ideal triangles goes to infinity, a generic hyperbolic surface in Brooks-Makover’s model has Cheeger constant less than $\frac{3}{2\pi }+ϵ$.