arXiv subjects

Ellen Krusell

Publications and source records attributed to Ellen Krusell.

3 recordsLinked to original sources

arXiv2025

Polyakov-Alvarez Formula for Curvilinear Polygonal Domains with Slits

We consider the $\zeta$-regularized determinant of the Friedrichs extension of the Dirichlet Laplace-Beltrami operator on curvilinear polygonal domains with corners of arbitrary positive angles. In particular, this includes slit domains. We obtain a short time asymptotic expansion of the heat trace using a classical patchwork method. This allows us to define the $\zeta$-regularized determinant of the Laplacian and prove a comparison formula of Polyakov-Alvarez type for a smooth and conformal change of metric.

math-ph↗

arXiv2024

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

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.

math.CV↗

arXiv2024

Commutation relations for two-sided radial SLE

We study the commutation relation for 2-radial SLE in the unit disc starting from two boundary points. We follow the framework introduced by Dub\'{e}dat. Under an additional requirement of the interchangeability of the two curves, we classify all locally commuting 2-radial SLE$_\kappa$ for $\kappa\in (0,8)$: it is either a two-sided radial SLE$_\kappa$ with spiral of constant spiraling rate or a chordal SLE$_\kappa$ weighted by a power of the conformal radius of its complement. Namely, for fixed $\kappa$ and starting points, we have exactly two one-parameter continuous families of locally commuting 2-radial SLE. Two-sided radial SLE with spiral is a generalization of two-sided radial SLE (without spiral) and satisfies the resampling property. We also discuss the semiclassical limit of the commutation relation as $\kappa \to 0$. In particular, we show that the limit for the second family with an appropriately chosen power of conformal radius is a chord that minimizes a modified chordal Loewner energy, which is unique only when the endpoints are not antipodal.

math.PR↗