Spectral Structure of the Mixed Hessian of the Dispersionless Toda $τ$-Function

Abstract

We analyze the mixed Hessian of the dispersionless Toda $τ$-function for the $s$-fold symmetric one-harmonic polynomial conformal map. The inverse branch exhibits two distinct thresholds: an analytic threshold $ζ_c$, where the dominant square-root singularity reaches the circle of convergence, and a later geometric threshold $ζ_{\mathrm{univ}}>ζ_c$, where the map ceases to be univalent. We prove that the first spectral instability occurs already at $ζ_c$. In each symmetry sector, the weighted subcritical realization has exactly one logarithmically diverging eigenvalue, whereas the remaining spectrum stays bounded and, after removal of the singular direction, converges to that of a compact limiting remainder. We further continue the corresponding scalar Gram functions beyond $ζ_c$, showing that they admit a generalized hypergeometric description, a Cauchy--Stieltjes representation, and, for $1\le p\le s$, a realization as Weyl functions of bounded Jacobi operators. In particular, these scalar quantities remain finite at $ζ_{\mathrm{univ}}$. This identifies analytic criticality, rather than loss of univalence, as the first spectral threshold of the Toda Hessian.

Figures (4)

• Figure 1: Logarithmic spectral asymptotics of the weighted Gram block $\widetilde{G}^{(q)}(\zeta)$ for $s=3,5$ in sector $q=1$ ($\beta=1$, $N=30$). Top row: the leading eigenvalue $\mu_1^{(q)}(\zeta)$ plotted against $L(\zeta)$, showing the asymptotically affine law $\mu_1^{(q)}(\zeta)=\Gamma^{(q)}\,L(\zeta)+O(1)$; the dashed line is a tail linear fit. Bottom row: the normalized eigenvalues $\mu_k^{(q)}(\zeta)/L(\zeta)$. The ratio $\mu_1^{(q)}(\zeta)/L(\zeta)$ approaches a positive constant, whereas $\mu_k^{(q)}(\zeta)/L(\zeta)\to0$ for $k\ge2$.
• Figure 2: Soft spectrum after removal of the stiff direction for $s=3,5$ ($\beta=1$, $N=40$). Top row: the soft branches $\mu_2^{(q)}(\zeta),\dots,\mu_6^{(q)}(\zeta)$ in sector $q=1$, plotted against $1/L(\zeta)$. Their flattening as $1/L(\zeta)\to0$ shows that the soft branches remain bounded as $\zeta\uparrow\zeta_c$. Bottom row: finite-$\zeta$ snapshots of the soft branches for all sectors $q=1,\dots,s$ at $\zeta/\zeta_c=0.9999$.
• Figure 3: Continued Gram weight $\sigma_p^{\mathrm{cont}}(u)$ and discontinuity density $\rho_p(u)$ for $s=3,5$. Top row: the analytically continued Gram weight $\sigma_p^{\mathrm{cont}}(u)$ on the subcritical side $0<u<\zeta_c^2$. Bottom row: the discontinuity density $\rho_p(u)$ on the supercritical side $u>\zeta_c^2$. The open circles at $u=\zeta_c^2$ mark the edge values $\rho_p(\zeta_c^2)>0$. For larger values of $p$, the density becomes negative away from the edge, showing that positivity at $u=\zeta_c^2$ does not persist on the entire supercritical branch.
• Figure 4: Structure of the first soft modes of $\widetilde{C}^{(q)}_{*,\perp}$ for $s=3,5$, with $q=1$, $\beta=1$, and $N=40$. The curves show the signed components of the eigenvectors $\phi_2^{(q)},\phi_3^{(q)},\phi_4^{(q)},\phi_5^{(q)}$, corresponding to the soft eigenvalues $\mu_{2,*}^{(q)},\mu_{3,*}^{(q)},\mu_{4,*}^{(q)},\mu_{5,*}^{(q)}$. As the mode index increases, the eigenvectors become more oscillatory while remaining concentrated near low values of the lattice index $p_j=q+js$.

Theorems & Definitions (110)

• Theorem : A: Rank-one logarithmic instability
• Remark 1.1: Contrast with the Grunsky operator
• Theorem : B: Analytic continuation of the scalar Gram data
• Proposition : C: Separation of thresholds
• Proposition 2.1: Functional equation for the inverse map
• proof
• Lemma 2.2: Critical point and local expansion
• proof
• Remark 2.3: Geometry in the $z$-plane
• Proposition 2.4: Raney numbers