Boundedness criteria for real quivers of rank 3

We study the boundedness of a mutation class for quivers with real weights. The main result is a characterization of bounded mutation classes for real quivers of rank 3.

💡 Research Summary

The paper investigates the boundedness of mutation classes of rank‑3 quivers whose edges carry real (non‑negative) weights. A mutation class 𝓡(Q) is called bounded if the supremum of the absolute values of all weights appearing in any quiver of the class is finite. For integer‑weighted quivers boundedness coincides with mutation‑finiteness, but for real‑weighted quivers the two notions diverge: there exist infinite mutation classes that are nevertheless bounded.

The authors’ main contribution is a complete, easily checkable criterion for boundedness in the rank‑3 case. Writing a quiver as Q = (p,q,r) with p ≥ q ≥ r > 0, they introduce the Markov constant

 C(p,q,r) = p² + q² + r² − pqr (if Q is cyclic)

or

 C(p,q,r) = p² + q² + r² + pqr (if Q is acyclic).

Theorem 1.1 states that the mutation class 𝓡(Q) is bounded if and only if two simple inequalities hold:

1. p ≤ 2, and
2. C(p,q,r) ≤ 4.

Thus the entire boundedness problem reduces to evaluating the largest weight and a single algebraic expression.

Two independent proofs are provided.

First proof (analytic).
The authors develop explicit recurrence relations for the weights under successive mutations. Lemma 2.3 shows that if a cyclic quiver has C > 4 then its largest weight must exceed 2. Proposition 2.2 proves that any unbounded class must contain a cyclic quiver with a weight larger than 2. By constructing a mutation sequence that alternates at vertices 1 and 2, they demonstrate that when either p > 2 or C > 4 the weights q_i, r_i grow without bound (essentially geometrically). Conversely, when p ≤ 2 and C ≤ 4 they establish uniform upper bounds using elementary inequalities (e.g., inequality (4) in the text). This approach is elementary, self‑contained, and yields explicit numerical bounds.

Second proof (geometric).
The alternative argument relies on the geometric model of rank‑3 mutation classes introduced by Felikson and Tumarkin. In that framework each mutation class corresponds to a tiling of a constant‑curvature surface: spherical (positive curvature), Euclidean (zero curvature), or hyperbolic (negative curvature). The Markov constant C determines the curvature: C ≤ 4 places the class in the spherical/Euclidean regime, where the associated triangle has side lengths bounded by 2, guaranteeing bounded weights. When C > 4 the class lives in hyperbolic space, where repeated mutations correspond to moving deeper into the hyperbolic plane, causing the side lengths (i.e., quiver weights) to diverge. The condition p ≤ 2 emerges naturally from the requirement that a spherical or Euclidean triangle with side lengths p,q,r exists; if p > 2 such a triangle cannot be realized, forcing the class into the hyperbolic (unbounded) regime. This proof provides a conceptual, visual understanding of why the algebraic inequalities are the exact threshold.

The paper also discusses the relationship with mutation‑finite quivers. Existing results (e.g., Felikson‑Tumarkin’s classification) show that mutation‑finite rank‑3 quivers are precisely those with C ≤ 4; the present work extends this by showing that even among mutation‑infinite quivers, boundedness occurs exactly when the same inequality holds together with p ≤ 2.

To illustrate the theory, the authors supply a Python implementation that generates random mutation sequences and visualizes the evolution of the weights (Figures 1 and 2). These examples demonstrate that quivers satisfying the criterion produce bounded trajectories, while those violating either condition exhibit weight blow‑up.

In conclusion, the paper delivers a sharp, computationally trivial test for boundedness of rank‑3 real‑weighted quivers, proves it by two complementary methods, and situates the result within the broader landscape of quiver mutation dynamics and cluster algebra geometry. This criterion is expected to be useful for researchers studying dynamical properties of quiver mutations, geometric realizations of cluster algebras, and related areas such as Teichmüller theory and discrete integrable systems.