Local Rank-One Logarithmic Instability for the Mixed Hessian of the Dispersionless Toda $τ$-Function

Abstract

We study the mixed Hessian of the dispersionless Toda $τ$-function for polynomial conformal maps, in scale-free parameters. An explicit logarithmic-kernel identity yields a symmetry-block decomposition of the Hessian, while the analytic input is the inverse-map generating function $U(x;ζ)$ and its first dominant singularities. Along a transversal subcritical path approaching a simple critical point, assume that the Taylor branch of $U$ has a unique dominant $s$-orbit of simple square-root branch points and satisfies a uniform continuation hypothesis. Then, for a fixed weighted renormalization of each symmetry block, we prove a local rank-one logarithmic instability: exactly one variational eigenvalue diverges logarithmically, while all higher variational eigenvalues remain bounded. After subtraction of the singular rank-one term, the remainder converges in operator norm to a compact limit. Along Laplacian growth, if the critical crossing occurs before loss of univalence, the same spectral transition precedes geometric breakdown. The result is local rather than global: it isolates the mechanism responsible for the first instability and records an abstract criterion for extension beyond polynomial leaves.

Figures (3)

• Figure 1: Numerical behavior consistent with Theorem \ref{['thm:universality']} on the two--harmonic leaf $U=1+\zeta_1 x^3U^3+\zeta_2 x^6U^6$. We fix $\zeta_2=0.01$ and approach the critical locus along the transversal slice $\zeta_1\uparrow \zeta_{1,c}(\zeta_2)$, with $\delta:=1-\zeta_1/\zeta_{1,c}(\zeta_2)$. The spectra are computed from a truncation of the renormalized Gram blocks $\widetilde{G}^{(q)}$ with $J=70$, $\alpha=2$, and $\beta=1$. Top row: the leading eigenvalue $\mu_1^{(q)}$ for the representative blocks $q=1,2$, plotted against $\log(1/\delta)$, together with linear tail fits. Bottom row: the normalized eigenvalues $\mu_k^{(q)}/\log(1/\delta)$, $k=1,\dots,5$, on a logarithmic vertical scale. The plots show one logarithmically diverging stiff mode, while the normalized soft modes tend to zero.
• Figure 2: Characteristic boundaries for two explicit $N=\infty$ leaves.(a) Single pole. On the real slice $b\in(-1,1)$, the level $\rho_{\mathrm{char}}=1$ is given by the two explicit curves $b+2\sqrt c=1$ and $b-2\sqrt c=-1$, meeting at $(b,c)=(0,1/4)$. By Proposition \ref{['prop:single-pole-rho-star']}, this is also the exact phase boundary for the true analyticity radius $\rho_*$. (b) Single log. Principal-sheet level curves of the characteristic modulus $\rho_{\mathrm{char}}(b,\gamma)$. The thick curve is the level $\rho_{\mathrm{char}}=1$. By Remark \ref{['rem:single-log-principal-sheet']}, this should be read only as a principal-sheet characteristic boundary supported by numerical continuation, not as a closed-form theorem for $\rho_*$.
• Figure 3: Single-log leaf: discriminant transition on the principal sheet.(a) Representative slice $\gamma=0.05$. The vertical line marks the discriminant point $b_{\mathrm{disc}}=4\gamma$. The solid curve is the active characteristic modulus $\rho_{\mathrm{char}}=\min\{|x_*^\pm|\}$, the dotted curve is the complementary branch, and the horizontal line is the level $\rho_{\mathrm{char}}=1$. (b) Representative slices for several values of $\gamma$. The change of shape at the marked points is caused by branch splitting at the discriminant line and does not, by itself, signal characteristic criticality.

Theorems & Definitions (85)

• Remark 2.1: Infinite--dimensional families
• Proposition 2.2: Inverse map equation
• proof
• Lemma 2.3: $s$--symmetry
• proof
• Definition 2.4: Radius of analyticity
• Remark 2.5: Taylor branch and chosen Taylor sheet
• Proposition 2.6: Kernel identity for the mixed Toda Hessian
• Definition 2.7: Scale--invariant mixed Hessian
• Remark 2.8: One--mode case and Raney numbers