Many-point tropical relaxation and the Monge--Amp\`ere equation

Abstract

We prove a quantitative tropical approximation to the planar Aleksandrov Monge--Amp\`ere equation. Let $\Omega\subset\mathbb R^2$ be a bounded open convex domain, fix $K\Subset\Omega$, and let $F_N=G_{P_N}(0_\Omega)$ be the minimal nonnegative concave tropical series with integral slopes, zero boundary values, and corner locus containing a universally generic $N$-point set $P_N\subset K$. Set $u_N=N^{-1/2}F_N$ and $\mu_N=N^{-1}\sum_{p\in P_N}\delta_p$. For every compact $L\Subset\Omega$ we prove $\left|\int\varphi\,d(\mathrm{MA}(u_N)-\mu_N)\right|\le C(\Omega,K,L)N^{-1/2}(\|\varphi\|_\infty+\|\nabla\varphi\|_\infty)$ for $\varphi\in C_c^1(\Omega)$ supported in $L$. If $\mu_N\rightharpoonup\mu$, where $\mu$ is a probability measure supported in $K$, then $u_N$ converges uniformly on $\overline\Omega$ to the unique continuous concave zero-boundary Aleksandrov solution of $\mathrm{MA}(F)=\mu$, and $\mathrm{MA}(u_N)\rightharpoonup\mu$ vaguely in $\Omega$. No regularity or strict convexity of $\partial\Omega$ is assumed. For bounded rational convex polygons, strong genericity suffices. If $P\subset K$ is strongly generic with $|P|=N$ and $F_P=G_P(0_\Omega)$, its tropical curve has exactly $N$ bounded cells; the duals of the uncut marked carriers form a spanning tree; every compact internal edge has weight one; and $\mathrm{MA}(F_P)(\Omega^\circ)=N-1+\tfrac12D_{\mathrm{term}}(F_P)$, with $D_{\mathrm{term}}(F_P)=O_{\Omega,K}(\sqrt N)$. For strongly generic sequences satisfying the same empirical-measure hypothesis, the normalized curvature measures converge weakly on the closed polygon. We also obtain almost-sure limits for i.i.d. samples from absolutely continuous laws supported in $K$, affine covariance of the continuum solution, and, for source sequences covered by the polygonal theorem, a configuration-dependent Abelian-sandpile diagonal.