Concentration conjecture for separating systoles of random triangulations

Fix $\theta\in(0,1/2)$ and a sequence $(g_n)$ such that $g_n/n\rightarrow\theta$. Let $\mathbf{T}_{2n,g_n}$ be a uniform triangulation of genus $g_n$ with $2n$ faces. The separating systole is the length of the shortest separating non-contractible cycle, and the simple separating systole is the length of the shortest simple separating non-contractible cycle; denote them by $\operatorname{sepsys}(\mathbf{T}_{2n,g_n})$ and $\operatorname{sepsys}(\mathbf{T}_{2n,g_n})^*$, respectively. Separating systole concentration conjecture. For every $\theta\in(0,1/2)$, there exist constants $0<S(\theta)<S^*(\theta)$ such that

$$\frac{\operatorname{sepsys}(\mathbf{T}_{2n,g_n})}{\log(n)}\rightarrow S(\theta)\quad\text{and}\quad\frac{\operatorname{sepsys}(\mathbf{T}_{2n,g_n})^*}{\log(n)}\rightarrow S^*(\theta)$$

in probability as $n\rightarrow\infty$.

The preceding result gives only logarithmic-order bounds for the separating systole. This conjecture predicts limiting constants and that the simple separating systole is strictly larger asymptotically.