Many-point tropical relaxation and the Monge--Ampère equation

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Original abstract

We construct an incidence-driven tropical approximation of the planar Aleksandrov Monge--Ampère equation. Let $Ω\subset\mathbb R^2$ be a bounded open convex domain, let $K\SubsetΩ$, and let $F_N=G_{P_N}0_Ω$ be the minimal nonnegative concave tropical series with integral slopes, zero boundary values, and corner locus containing an $N$-point set $P_N\subset K$. For universally generic configurations whose empirical measures converge to $μ$, $N^{-1/2}F_N\longrightarrow F_{μ,Ω}$ uniformly on $\overlineΩ$, where $F_{μ,Ω}$ is the unique continuous concave Aleksandrov solution of $\mathrm{MA}(F)=μ$ 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 $\partialΩ$. 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 $\mathrm{MA}(F_N)(Ω^\circ)=N-1+\frac{1}{2}D_{\mathrm{term}}(F_N)$, with $D_{\mathrm{term}}(F_N)=O(\sqrt N)$. 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.

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