Iterated Gilbert Mosaics and Poisson Tropical Plane Curves

TL;DR

This paper introduces an iterated Gilbert model that converges to a Poisson line process and applies this to derive scaling limits for Poisson tropical plane curves, bridging stochastic and tropical geometry.

Contribution

It develops an iterated Gilbert model and proves its convergence to a Poisson line process, enabling analysis of tropical plane curves in a probabilistic framework.

Findings

Convergence of iterated Gilbert mosaics to Poisson line process

Derivation of scaling limits for Poisson tropical plane curves

Establishment of connections between stochastic and tropical geometry

Abstract

We propose an iterated version of the Gilbert model, which results in a sequence of random mosaics of the plane. We prove that under appropriate scaling, this sequence of mosaics converges to that obtained by a classical Poisson line process with explicit cylindrical measure. Our model arises from considerations on tropical plane curves, which are zeros of random tropical polynomials in two variables. In particular, the iterated Gilbert model convergence allows one to derive a scaling limit for Poisson tropical plane curves. Our work raises a number of open questions at the intersection of stochastic and tropical geometry.