Growth constant for meandric systems with a component weight

Determine the growth constant g_1(q) governing the asymptotic enumeration of meandric systems weighted by q per connected component for 0\leq q\leq 2, for which the paper states a conjectural asymptotic form and leaves g_1(q) unknown.

Background

The paper studies meandric systems with a weight q assigned to each connected component. For 0\leq q\leq 2, it reports a conjectured and numerically verified asymptotic form for the coefficients m_n{(1)}(q), involving a q-dependent exponential growth constant g_1(q) and a q-dependent susceptibility exponent.

The growth constant is not given in closed form in the paper. The authors estimate it numerically from finite coefficient data, so determining g_1(q) remains an unresolved quantitative problem within this family of weighted meandric systems.

References

for some unknown $g_1(q)$.

— Liouville Quantum Duality and Random Planar Maps  (2507.12203 - Duplantier et al., 16 Jul 2025) in Section 10.2, “Meandric systems weighted by both connected components and irreducible blocks” (Section \ref{sec:weightq})

It is conjectured that $$ m_n\sim C_{\text{even}} Rn n{-\alpha} \quad \text{for even } n, \qquad m_n\sim C_{\text{odd}} Rn n{-\alpha} \quad \text{for odd } n, $$ as $n\to\infty$, where $C_{\text{even}}$ and $C_{\text{odd}}$ are positive constants and $\alpha=\frac{29+\sqrt{145}}{12}$.

— S-meandric Permutations and Tangency Polynomials  (2609.30068 - Belousov et al., 24 Sep 2026) in Section 6, Asymptotics and enumeration