2026/07/28 by Nikita Kalinin, Ernesto Lupercio, Higinio Serrano +1
Mathematics · #math.AP #math.AG #math.MG
We construct an incidence-driven tropical approximation of the planar Aleksandrov Monge--Ampère equation. Let Ω⊂\mathbb R2 be a bounded open convex domain, let K\SubsetΩ, and let FN=GPN0Ω be the minimal nonnegative concave tropical series with integral slopes, zero boundary values, and corner locus containing an N-point set PN⊂ K. For universally generic configurations whose empirical measures converge to μ, N-1/2FN\longrightarrow Fμ,Ω uniformly on Ω, where Fμ,Ω is the unique continuous concave Aleksandrov solution of MA(F)=μ with zero boundary values. On every compact L\SubsetΩ we prove an O(N-1/2) bounded-Lipschitz-type estimate for the curvature discrepancy. No regularity or strict-convexity assumption is imposed on ∂Ω. For rational polygons, strong genericity holds on an open dense full-measure locus. The tropical curve has exactly N bounded cells, the marked dual edges form a spanning tree, and every compact internal edge has weight one. These finite statements yield global weak curvature convergence and the exact identity MA(FN)(Ω^∘)=N-1+(1)/(2)Dterm(FN), with Dterm(FN)=O(√ N). The proof combines tropical interpolation, semilinear marked topology, an Euler--Pick curvature formula, minimal coefficient deformations, tangential coarea, and a weighted Crofton estimate uniform over rational polygonal exhaustions. We also obtain almost-sure limits for random point clouds, full affine covariance of the continuum solution, and a configuration-dependent Abelian-sandpile diagonal.