Planar tropical caustics: trivalency and convexity

TL;DR

This paper investigates planar tropical caustics $\mathcal{K}_\Phi$ for convex domains $\Phi\subset\mathbb{R}^2$, establishing a graphical proof that all intermediate vertices are trivalent. It relates the caustics to tropical wave fronts, $A_n$-type singularities, and toric-symplectic data via the propagation of $\Phi(t)$ and the evolution of the associated toric surface $S(t)$, then analyzes several explicit examples (e.g., complements to amoebas of a line, ellipses, nodal cubic). Beyond convex domains, it outlines three strategies—via symplectic geometry of toric surfaces, lattice-based approximations, and caustic-coordinate extensions—for defining and understanding tropical caustics, along with a critical comparison of their implications. The work suggests that caustics encode domain invariants and may permit a broader, higher-dimensional tropical theory with links to moduli of quantum toric varieties.

Abstract

Tropical caustic of a convex domain on the plane is a canonically associated tropical analytic curve inside the domain. In this note we give a graphical proof for the classification of its intermediate vertices, implying in particular that they are always trivalent. Apart from that we explain how various known examples of tropical caustics are constructed and discuss the possibility of relaxing the convexity condition for the domain.

Figures (21)

• Figure 1: Right: the result of dropping a single grain to the center of a disc on the maximal stable state of the sandpile model. Left: tropical caustic of the disc.
• Figure 2: The three fundamental entities of tropical optics. Each one uniquely restores the other two.
• Figure 3: Possible schemes of collisions in the particle process (defined in Section \ref{['sec_trival']}) at the final time in the case when a mass two particle is formed, its trajectory is traced in bold. The configuration on the right is realizable for all natural $n.$ Below: dual polygons of corresponding vertices where collisions take place. Note that, in this case, there is a second collision with another weight two particle formed at the same (final) time. These two particles move towards each other along the final locus $\Phi(t_\Phi)$. Each pair of the schemes above can occur independently at the two sides of this edge, if the domain $\Phi$ is compact. For a non-compact domain with parallel rational asymptotes of the boundary) the final locus is a ray, i.e. the caustic final singularity type is determined by a single picture out of these types.
• Figure 4: Sixteen types of lattice polygons (up to automorphisms of the lattice) with a single lattice point in the interior together with their tropical caustics. Each caustic consists of segments connecting the central point to the vertices. The polygons are paired by duality, the weight of an edge of the caustic is the length of the corresponding side of the dual polygon. Note that some of these types are self-dual, in a sense of identifying the lattices $M$ and $N.$ Observe that the sums of lattice perimeters of two dual polygons are always equal to 12 and that the self-dual types are exactly the 3-,4-,5- and 6-gon with the perimeter 6. This is an instance of a general Noether-type formula established in MS23 equating the sum of the perimeter of the domain boundary and its caustic to the length of the final segment (multiplied by $4$), plus the final time $t_\Phi$ (multiplied by $12$). These sixteen types classify possible final singularities of tropical caustics in the case when the final locus $\Phi(t_\Phi)$ of the wave front propagation is a point.
• Figure 5: The rule for constructing a caustic of a cone. One should take the convex hull of the set of non-zero lattice points in the dual cone. The finite edges in the resulting polygonal domain are in one-to-one correspondence (via orthogonality) with the rays in the caustic of the original cone, with the length of a segment being equal to the weight of the ray.

Theorems & Definitions (14)

• Remark 1
• Proposition 2
• Proposition 3
• Theorem 4: Huygens' principle
• Proposition 5
• Proposition 6
• Proposition 7
• Theorem 8
• Theorem 9
• Theorem 10