2024/10/11 by Ellen Krusell, Krusell, Ellen · 2 citations
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Probability (math.PR) #Spectral Theory in Mathematical Physics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2410.08969
openalex publication_date 2024/10/11 · openalex created_date 2024/10/16 · openalex updated_date 2026/07/28
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 SLEκ(ρ), as κ→ 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 SLE0(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.