General tropical convergence of harmonic amoebas

TL;DR

This paper studies how harmonic amoebas of degenerating Riemann surfaces converge to tropical curves, extending previous results to more complex cases and aiming to impact moduli space compactification and dimer model analysis.

Contribution

It extends Lang's simple tropical convergence results to non-simple cases using Schottky uniformization, broadening understanding of harmonic amoeba convergence.

Findings

Describes tropical convergence of harmonic amoebas to non-simple tropical curves.

Extends Lang's results on tropical convergence beyond simple cases.

Provides insights for compactifying moduli spaces and analyzing dimer models.

Abstract

By using Schottky uniformization theory of degenerating algebraic curves, we describe the tropical convergence of harmonic amoebas of pointed Riemann surfaces to tropical curves which are not necessarily simple. We extend Lang's results on the simple tropical convergence based on the Frenchel-Nielsen coordinates to the nonsimple case. Our results are hoped to give contributions in compactifying the moduli space of pointed Riemann surfaces with tropical curves, and in studying crystallization of general dimer models.