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

Abstract

We prove a quantitative tropical approximation to the planar Aleksandrov Monge--Ampère equation. Let $\Omega \subset {\mathbb{R}}^2$ be a bounded open convex domain, fix $K⋐\Omega$, 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 $L⋐\Omega$ we prove $\left|\int \phi d(\mathrm{MA}(u_N)-{\mu }_N)\right|\le C(\Omega ,K,L)N^{-1/2}(\Vert \phi {\Vert }_{\infty }+\Vert \nabla \phi {\Vert }_{\infty })$ for $\phi \in C_c^1(\Omega )$ 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 $\mathrm{MA}(F_P)({\Omega }^{\circ })=N-1+\frac{1}{2}{D}_{\mathrm{term}}(F_P)$, 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.

67 pages, 3 figures. Revised exposition and proof clarifications; corrected numerical appendix. Main results unchanged