The limiting density conjecture for rays of random toric surfaces

Let $S_{\geq k}$ be the subset of rays of the complete fan associated with the smooth case whose corresponding exceptional divisors have appropriate negative self-intersection number, and let $\Sigma_h(1)$ denote the set of rays of the fan $\Sigma_h$. For $n$, let $T_n$ be the $n$-th triangular number:

$$T_n=\frac{n^2+n}{2}.$$

The limiting density conjecture. For $k>1$,

$$\lim_{h\rightarrow \infty} \frac{\left|S_{\geq k}\right|}{\left|\Sigma_h(1)\right|}=\frac{2}{T_k}.$$

This conjecture refines the bounds obtained for the density of rays in the complete fan arising in the smooth case. Numerical data support the proposed limiting values, and the result would in particular imply that both $S_{\geq k}$ and the set $S_k$ of rays with the corresponding exact index have positive density.