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Multi-time Loewner energy: rate function for large deviation

Mo Chen, Chongzhi Huang, Hao Wu

Abstract

The classification of probability measures that satisfy both conformal invariance and domain Markov property is equivalent to characterizing solutions to the Belavin--Polyakov--Zamolodchikov (BPZ) equations, as established by Dubédat~[Dub07]. In this context, the partition functions for half-watermelon SLE and for multi-radial SLE serve as fundamental solutions to the BPZ equations. In this article, we investigate the large deviation principle for both half-watermelon SLE and multi-radial SLE. The associated rate function is given by the multi-time Loewner energy, introduced in~[CHPW26]. As applications, we provide an alternative proof of the large deviation principle for Dyson Brownian motion, as well as a new derivation of the boundary perturbation property of the multi-time Loewner energy.

Multi-time Loewner energy: rate function for large deviation

Abstract

The classification of probability measures that satisfy both conformal invariance and domain Markov property is equivalent to characterizing solutions to the Belavin--Polyakov--Zamolodchikov (BPZ) equations, as established by Dubédat~[Dub07]. In this context, the partition functions for half-watermelon SLE and for multi-radial SLE serve as fundamental solutions to the BPZ equations. In this article, we investigate the large deviation principle for both half-watermelon SLE and multi-radial SLE. The associated rate function is given by the multi-time Loewner energy, introduced in~[CHPW26]. As applications, we provide an alternative proof of the large deviation principle for Dyson Brownian motion, as well as a new derivation of the boundary perturbation property of the multi-time Loewner energy.
Paper Structure (36 sections, 54 theorems, 285 equations, 2 figures)

This paper contains 36 sections, 54 theorems, 285 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Large deviation for half-watermelon SLE
  3. Curve Spaces.
  4. Large deviation for multi-radial SLE with spiral
  5. Curve Spaces.
  6. Large deviation for Dyson Brownian motion
  7. Boundary perturbation of the energy
  8. Preliminaries
  9. Poisson kernels and Brownian measures
  10. Estimates on Bessel-type processes
  11. Return estimate for chordal SLE with force points
  12. Chordal Loewner chain and its energy
  13. A monotone coupling
  14. Proof of Proposition \ref{['prop::SLE_return_chordal']}
  15. Estimate on the bad event
  16. ...and 21 more sections

Key Result

Theorem 1.2

Fix $n\ge 1$ and $(n+1)$-polygon $(\Omega; \boldsymbol{x}, y)$. The family $\{ \mathbb{P}_{_{n}}^{(\kappa)}(\Omega; \boldsymbol{x}, y)\}_{\kappa\in (0,4]}$ of laws of half-$n$-watermelon $\mathrm{SLE}_{\kappa}$ satisfies large deviation principle in the space $(\mathfrak{X}_{_{n}}(\Omega; \boldsymbo

Figures (2)

  • Figure 1.1: Examples of the curve spaces.
  • Figure 1.2: Simulations of Dyson Brownian motion and Dyson circular ensemble.

Theorems & Definitions (97)

  • Definition 1.1: Half-watermelon SLE
  • Theorem 1.2
  • Definition 1.3: Multi-radial SLE with spiral
  • Theorem 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['lem::chordalBessel_estimate']} (1)
  • ...and 87 more