Spectral Response Geometry for Liouville Brownian Motion

Abstract

For Dirichlet Liouville Brownian motion associated with a subcritical Gaussian multiplicative chaos measure on a bounded planar domain, we study the response of finite spectral blocks under Cameron--Martin perturbations of the underlying Gaussian free field. For each fixed $0<γ<2$, we prove that, almost surely, all unordered eigenfunction-product measures in every finite simple spectral block are linearly independent. This strengthens the corresponding diagonal independence statement and implies surjectivity of the joint eigenvalue and eigenframe response map. We use this transversality to construct real-analytic local deformations that preserve a prescribed finite block of eigenvalues while varying its eigenframe. We then identify the least Cameron--Martin energy required to realize a given infinitesimal angular response, expressing it through a positive Schur complement of the response operator. Finally, on finite-dimensional Gaussian slices, we derive the conditional law associated with the local isospectral leaf and compare the induced slice metric with the full Cameron--Martin minimum. The results provide a local geometric description of spectral flexibility for Liouville Brownian motion.