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

We construct an incidence-driven tropical approximation of the planar Aleksandrov Monge–Ampère equation. Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be a bounded open convex domain, let $K⋐\mathrm{\Omega }$, and let $F_N=G_{P_N}{0}_{\mathrm{\Omega }}$ 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 $\mu$, $N^{-1\mathrm{/}2}{F}_N⟶F_{\mu ,\mathrm{\Omega }}$ uniformly on $\bar{\mathrm{\Omega }}$, where $F_{\mu ,\mathrm{\Omega }}$ is the unique continuous concave Aleksandrov solution of $\mathrm{M}\mathrm{A}(F)=\mu$ with zero boundary values. On every compact $L⋐\mathrm{\Omega }$ we prove an $O(N^{-1\mathrm{/}2})$ bounded-Lipschitz-type estimate for the curvature discrepancy. No regularity or strict-convexity assumption is imposed on $\mathrm{\partial }\mathrm{\Omega }$. 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 $\mathrm{M}\mathrm{A}(F_N)({\mathrm{\Omega }}^{\circ })=N-1+\frac{1}{2}{D}_{\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}}(F_N)$, with $D_{\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}}(F_N)=O(\sqrt{N})$. 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.