Tropicalization and cluster asymptotic phenomenon of generalized Markov equations

TL;DR

This paper explores the tropicalization of generalized Markov equations, revealing their structural similarity to classical Euclid trees and demonstrating an asymptotic relationship through cluster mutations, with implications for conjectures on uniqueness and rationality.

Contribution

It introduces a deformed tropicalization framework for generalized Markov equations and establishes their asymptotic connection to classical Euclid trees using cluster mutation techniques.

Findings

Tropicalized tree structure matches classical Euclid tree.

Generalized Euclid tree converges to classical Euclid tree up to scalar.

Asymptotic relationship between generalized Markov and Euclid trees demonstrated.

Abstract

The generalized Markov equations are deeply connected with the generalized cluster algebras of Markov type. We construct a deformed Fock-Goncharov tropicalization for the generalized Markov equations and prove that their tropicalized tree structure is essentially the same as that of the classical Euclid tree. We then define the generalized Euclid tree and prove that it converges to the classical Euclid tree up to a scalar multiple. Moreover, by means of cluster mutations, we exhibit an asymptotic phenomenon, up to some limit q, between the logarithmic generalized Markov tree and the classical Euclid tree. A rationality conjecture of q is then put forward. We also propose a generalized Markov uniqueness conjecture for the generalized Markov equations, which illustrates an application of the asymptotic phenomenon.