# Asymptotic sign coherence conjecture

## TL;DR

This paper introduces the asymptotic sign coherence conjecture for c-vectors in general cluster algebras, proposing that under certain mutation sequences, these vectors become sign coherent, and proves it in specific cases.

## Contribution

It formulates a new conjecture on the asymptotic behavior of c-vectors in general cluster algebras and proves it for certain infinite rank 2 cases and a specific mutation sequence.

## Key findings

- Proved the conjecture for rank 2 infinite type cluster algebras.
- Validated the conjecture for a mutation sequence in the Markov quiver case.

## Abstract

The sign coherence phenomenon is an important feature of c-vectors in cluster algebras with principal coefficients. In this note, we consider a more general version of c-vectors defined for arbitrary cluster algebras of geometric type and formulate a conjecture describing their asymptotic behavior. This conjecture, which is called the asymptotic sign coherence conjecture, states that for any infinite sequence of matrix mutations that satisfies certain natural conditions, the corresponding c-vectors eventually become sign coherent. We prove this conjecture for rank 2 cluster algebras of infinite type and for a particular sequence of mutations in a cluster algebra associated with the Markov quiver.