Random-point homotopic curve shortening approximates affine curve-shortening flow

Let $\delta$ be an initial curve contained in a convex region $R$ of area $A$. For $t>0$, write $\delta'=\delta(t)$ for its affine curve-shortening flow whenever it is defined. For each fixed $n$, choose a set $P$ of $An^2$ obstacle points independently and uniformly at random from $R$, let $\gamma_0$ be the shortest curve homotopic to $\delta$ under the obstacle set $P$, and set

$$m=\lfloor c_{\mathrm r}tn^{4/3}\rfloor,$$

where $c_{\mathrm r}\approx 1.3$. Define $\gamma_m=\operatorname{\mathrm HCS}_P^{(m)}(\gamma_0)$. Random-point approximation conjecture. There exists a constant $c_{\mathrm r}\approx 1.3$ such that, as $n\to\infty$, the Fr\e9chet distance between $\gamma_m$ and $\delta'$ is almost surely smaller than some $\varepsilon=\varepsilon(n)$ tending to $0$. The claim extends the experimentally observed HCS–ACSF correspondence from uniform grids to random obstacle sets, with a different time-scaling constant; the source gives no resolution status.