Exact limiting distribution of normalized triangle shapes in the moduli space

Determine the limiting probability distribution on the moduli space of triangles \(\mathcal{M}_\Delta\) for the multiplicity-weighted set of similarity classes arising from lattice triangles with vertices in \(\mathbb{Z}^2\cap([-N,N]\times[-N,N])\), as \(N\to\infty\). Equivalently, determine the probability distribution of the normalized side-length triple \((a,b,c)\in\mathbb{R}^3_+\) obtained by dividing each side length by the semi-perimeter so that \(a+b+c=2\) when three points are chosen independently and uniformly at random in the unit square \([0,1]\times[0,1]\).

Background

The paper defines the moduli space of triangles $\mathcal{M}_\Delta$ via normalized side lengths $(a,b,c)$ with $a+b+c=2$, and studies the subset determined by lattice triangles. It proves density of lattice triangles in $\mathcal{M}_\Delta$ but shows that the similarity classes of lattice triangles with vertices in $[-N,N]^2$, counted with multiplicity, do not equidistribute with respect to the area measure.

Despite non-equidistribution, numerical evidence suggests the resulting distribution is close to uniform, motivating the problem of identifying the exact limiting distribution. The authors reformulate this limit in terms of the distribution of normalized side lengths when three points are chosen independently and uniformly at random in the unit square, which corresponds to the $N\to\infty$ scaling limit of lattice triangles.