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

Abstract

We study a weighted renormalization of the mixed Hessian of the dispersionless Toda $\tau$-function associated with polynomial conformal maps. The starting point is an explicit logarithmic-kernel representation, which yields a decomposition of the Hessian into symmetry blocks and reduces the spectral analysis to the inverse-map generating function $U(x;\zeta)$ and the geometry of its dominant singularities. Near a transversal subcritical approach to a simple analytic critical point, we identify a rank-one logarithmic spectral instability: in each renormalized symmetry block, exactly one variational eigenvalue diverges logarithmically, whereas the remaining variational eigenvalues stay bounded. The proof isolates the analytic mechanism behind this transition in the emergence of a dominant $s$-orbit of simple square-root branch points of the Taylor branch of the inverse map. We then apply the same framework to reduced Laplacian-growth trajectories and show that the same spectral transition occurs there under the same local continuation hypotheses. If, in addition, the reduced map remains univalent at the spectral crossing, then this transition occurs before geometric breakdown. The result is local and conditional: it identifies the mechanism of the first instability and formulates an abstract criterion for extensions beyond the polynomial class.