3 problems

Let $g_n$ satisfy … and let $\mathbf{T_{n,g_n}}$ be a uniform triangulation of genus $g_n$ with $3n$ edges. An induced subgraph is a $\kappa$-expander if every set of vertices has…

Let $q^{(n)}$ be the random uniform high-genus quadrangulation, and let $\operatorname{syst}(q^{(n)})$ denote the size of its shortest non-contractible cycle. Systole tightness con…

Let $q^{(n)}$ be the random uniform quadrangulation in the high-genus regime, with genus parameter determined by $\theta$, and let $\operatorname{diam}(q^{(n)})$ denote its graph d…