The $\ddot{I}$-Loewner Energy: Large Deviations, Minimizers, and Alternative Descriptions

Abstract

We introduce and study the $\ddot{I}$-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${}_{\hat{I}}º(\ddot{I})$, as $\hat{I}º\to 0+$ and $\ddot{I}>-2$ is fixed, with the $\ddot{I}$-Loewner energy as the rate function in both radial and chordal settings. The unique minimizer of the $\ddot{I}$-Loewner energy is the SLE${}_0(\ddot{I})$ curve. We show that it exhibits three phases as $\ddot{I}$ 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 $\ddot{I}$-Loewner energy in the reference point hitting phase, $\ddot{I}>-2$. In the radial setting we give an equivalent description in terms of the Dirichlet energy of $\log |h^{\prime }|$, 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 $\ddot{I}$-Loewner energy optimal way. Finally, we express the $\ddot{I}$-Loewner energy in terms of $\hat{I}\P$-regularized determinants of Laplacians.

59 pages, 6 figures