The $ρ$-Loewner Energy: Large Deviations, Minimizers, and Alternative Descriptions

Abstract: We introduce and study the $\rho$-Loewner energy, a variant of the Loewner energy with a force point on the boundary of the domain. We prove a large deviation principle for SLE$_\kappa(\rho)$, as $\kappa \to 0+$ and $\rho>-2$ is fixed, with the $\rho$-Loewner energy as the rate function in both radial and chordal settings. The unique minimizer of the $\rho$-Loewner energy is the SLE$_0(\rho)$ curve. We show that it exhibits three phases as $\rho$ varies and give a flow-line representation. We also define a whole-plane variant for which we explicitly describe the trace. We further obtain alternative formulas for the $\rho$-Loewner energy in the reference point hitting phase, $\rho > -2$. In the radial setting we give an equivalent description in terms of the Dirichlet energy of $\log|h'|$, where $h$ is a conformal map onto the complement of the curve, plus a point contribution from the tip of the curve. In the chordal setting, we derive a similar formula under the assumption that the chord ends in the $\rho$-Loewner energy optimal way. Finally, we express the $\rho$-Loewner energy in terms of $\zeta$-regularized determinants of Laplacians.