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

Ellen Krusell

Abstract

We introduce and study the $ρ$-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$_κ(ρ)$, as $κ\to 0+$ and $ρ>-2$ is fixed, with the $ρ$-Loewner energy as the rate function in both radial and chordal settings. The unique minimizer of the $ρ$-Loewner energy is the SLE$_0(ρ)$ curve. We show that it exhibits three phases as $ρ$ 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 $ρ$-Loewner energy in the reference point hitting phase, $ρ> -2$. 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 $ρ$-Loewner energy optimal way. Finally, we express the $ρ$-Loewner energy in terms of $ζ$-regularized determinants of Laplacians.

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

Abstract

We introduce and study the -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, as and is fixed, with the -Loewner energy as the rate function in both radial and chordal settings. The unique minimizer of the -Loewner energy is the SLE curve. We show that it exhibits three phases as 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 -Loewner energy in the reference point hitting phase, . In the radial setting we give an equivalent description in terms of the Dirichlet energy of , where 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 -Loewner energy optimal way. Finally, we express the -Loewner energy in terms of -regularized determinants of Laplacians.

Paper Structure

This paper contains 33 sections, 56 theorems, 363 equations, 6 figures.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Main results
  4. The $\rho$-Loewner energy
  5. Large deviation principle.
  6. Minimizers and flow-lines.
  7. Dirichlet energy formulas.
  8. $\zeta$-regularized determinants.
  9. Preliminaries
  10. The chordal and radial Loewner equations
  11. SLE processes
  12. Harmonic measure
  13. Chordal Loewner energy
  14. Large deviation principles
  15. -regularized determinants of Laplacians
  16. ...and 18 more sections

Key Result

Theorem 1

Fix $a\in\partial \mathbb D$, $b\in\overline \mathbb D\setminus \{a\}$, $c\in\partial\mathbb D\setminus\{a,b\}$, and $\rho>-2$. Let $\mathcal{X}^{(D;a,b)}$ denote the space of simple curves from $a$ to $b$ in $\mathbb D$ equipped with the Hausdorff topology. In this topology, the SLE$_\kappa(\rho)$

Figures (6)

  • Figure 1: Chordal SLE$_0(\rho)$ curves. On the left, the force point is on the boundary, and on the right, the force point is in the interior with $\arg z_0=\pi/2$.
  • Figure 2: Positively oriented whole-plane SLE$_0(\rho)$ curves started at $0$ in the direction $1$ with force point $0$ and reference point $\infty$. From left to right $\rho=-16,\ -8,\ -6,\ -5,\ -4,\ -3.5,\ -3,$ and $-2.5$. The origin is marked by a gray dot, and in the three right-most figures, the continuation of the flow-line, after self-intersection, is shown in gray. In particular, the SLE$_0(-6)$ is a circle through $0$ and SLE$_0(-4)$ is a cardioid with cusp at $0$.
  • Figure 3: This figure illustrates the set-up for the proof of Lemma \ref{['lemma:derivativeexists']}.
  • Figure 4: Illustration of the set-up in Lemma \ref{['lemma:conebound']}.
  • Figure 5: Illustration of the conformal maps $h_{T}$, $\tilde{h}_{\tilde{T}}$, and $h_{\tilde{\gamma}^0}$ from the proof of Theorem \ref{['thm:chordal']}.
  • ...and 1 more figures

Theorems & Definitions (132)

  • Definition 1: $\rho$-Loewner energy
  • Remark 1
  • Remark 2
  • Theorem 1: LDP
  • Remark 3
  • Remark 4
  • Theorem 2: Dirichlet energy formula, radial setting
  • Remark 5
  • Theorem 3: Dirichlet energy formula, chordal setting
  • Proposition 1: Determinants of Laplacians
  • ...and 122 more