Combinatorial mechanism for reproducing α-dependent leading asymptotics via Bessel expansions

Determine the combinatorial mechanism by which a direct approach that substitutes the small‑x asymptotic expansions of the Bessel functions J_ν(x) and Y_ν(x) into the Toeplitz determinant Δ_n(x,α) = det({C_{α/2−j+k}(x)}_{j,k=0}^{n−1}) with C_ν(x) = d1 J_ν(x) + d2 Y_ν(x), could reproduce the α‑dependent, piecewise leading asymptotic behaviors established for Δ_n(x,α) as x → 0 in Theorem 1.1.

Background

The paper derives small‑x asymptotics for Toeplitz determinants built from cylinder functions and, via a determinant formula, for special-function solutions of the Painlevé III equation. The results show that the leading power in the asymptotics depends piecewise on the parameter α, requiring a careful analytic approach using multiple contour integrals and asymptotic analysis.

The authors note that a naive attempt to insert Bessel function asymptotics directly into the determinant representation would need a combinatorial explanation for how different leading behaviors emerge as α varies. They explicitly state that this mechanism is unclear to them, highlighting an unresolved question about whether and how a purely combinatorial proof could reproduce the established α‑dependent leading terms.