Average harmonic spectrum of the whole-plane SLE

et al. 2017

Commun. Math. Phys.

Abstract. It has been shown that for f an instance of the whole-plane SLE unbounded conformal map from the unit disk D to the slit plane, the derivative moments E(|f (z)| p ) can be written in a closed form for certain values of p depending continuously on the SLE parameter κ ∈ (0, ∞). We generalize this property to the mixed moments, E |f (z)| p |f (z)| q , along integrability curves in the moment plane (p, q) ∈ R 2 depending continuously on κ, by extending the so-called Beliaev-Smirnov equation to this case. The generalization of this integrability property to the m-fold transform of f is also given. We define a novel generalized integral means spectrum, β(p, q; κ), corresponding to the singular behavior of the above mixed moments. By inversion, it allows a unified description of the unbounded interior and bounded exterior versions of whole-plane SLE, and of their m-fold generalizations. The average generalized spectrum of whole-plane SLE is found to take four possible forms, separated by five phase transition lines in the moment plane R 2 . The average generalized spectrum of the m-fold whole-plane SLE is directly obtained from the m = 1 case by a linear map acting in the moment plane. We also conjecture the precise form of the universal generalized integral means spectrum.Date: April 24, 2017. Key words and phrases. Whole-plane SLE, logarithmic coefficients, Beliaev-Smirnov equation, generalized integral means spectrum, universal spectrum.The first author wishes to thank the Isaac Newton Institute (INI) for Mathematical Sciences at Cambridge University, where part of this work was completed, for its hospitality and support during the 2015 program "Random Geometry", supported by EPSRC Grant Number EP/K032208/1. B.D. also gratefully acknowledges the support of a Simons Foundation fellowship at INI during the Random Geometry program. B.D. acknowledges financial support from the French Agence Nationale de la Recherche via the grant ANR-14-CE25-0014 "GRAAL"; he is also partially funded by the CNRS Projet international de coopération scientifique (PICS) "Conformal Liouville Quantum Gravity" n o PICS06769. The research by the second author is supported by a joint scholarship from MENESR and Région Centre; the third author is supported by a scholarship of the Government of Vietnam. B.D. and M.Z. are partially funded by the CNRS-insmi Groupement de Recherche (GDR 3475) "Analyse Multifractale". BD and MZ also gratefully acknowledge the continuous hospitality of the Institut Henri Poincaré in Paris. Figure 1. Loewner map z → f t (z) from D to the slit domain Ω t = C\γ([t, ∞)) (here slit by a single curve γ([t, ∞)) for SLE κ≤4 ). One has f t (0) = 0, ∀t ≥ 0. At t = 0, the driving function λ(0) = 1, so that the image of z = 1 is at the tip γ(0) = f 0 (1) of the curve. [From Ref. [14]].

mentioning

confidence: 94%

“…The fourth spectrum, β 1 (p, q), is the extension to non-vanishing q of a novel integral means spectrum, which was discovered and studied in Refs. [14] and [27,29], and which is due to the unboundedness of whole-plane SLE. As shown in Ref.…”

mentioning

confidence: 99%

Logarithmic Coefficients and Generalized Multifractality of Whole-Plane SLE

et al. 2017

Commun. Math. Phys.

“…The β 1 spectrum is the analogue for the exterior whole-plane SLE of the spectrum at infinity of Refs. [7,16] for the interior case (see also Ref. [5]).…”

Section: Figurementioning

confidence: 99%

“…Indeed, it was observed in Refs. [16,7] that the analysis provided in Ref. [2] is incomplete, and that the integral means spectrum could a priori be dominated by a novel spectrum, thought of as arising from the neighborhood of the starting point, and to be distinguished from that brought in by the vicinity of the tip.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Integral Means Spectrum of Whole-Plane SLE

Beliaev

Zinsmeister

2017

Commun. Math. Phys.

ABSTRACT. We complete the mathematical analysis of the fine structure of harmonic measure on SLE curves that was initiated in Ref.[2], as described by the averaged integral means spectrum. For the unbounded version of whole-plane SLE as studied in Refs. [7,16], a phase transition has been shown to occur for high enough moments from the bulk spectrum towards a novel spectrum related to the point at infinity. For the bounded version of whole-plane SLE of Ref.[2], a similar transition phenomenon, now associated with the SLE origin, is proved to exist for low enough moments, but we show that it is superseded by the earlier occurrence of the transition to the SLE tip spectrum.

The Coefficient Problem and Multifractality of Whole-Plane SLE & LLE

Nguyen

et al. 2014

Ann. Henri Poincaré

105

International audienceKarl Löwner (later known as Charles Loewner) introduced his famous differential equation in 1923 to solve the Bieberbach conjecture for series expansion coefficients of univalent analytic functions at level n = 3. His method was revived in 1999 by Oded Schramm when he introduced the Stochastic Loewner Evolution (SLE), a conformally invariant process which made it possible to prove many predictions from conformal field theory for critical planar models in statistical mechanics. The aim of this paper is to revisit the Bieberbach conjecture in the framework of SLE processes and, more generally, Lévy processes. The study of their unbounded whole-plane versions leads to a discrete series of exact results for the expectations of coefficients and their variances, and, more generally, for the derivative moments of some prescribed order p. These results are generalized to the “oddified” or m-fold conformal maps of whole-plane SLEs or Lévy–Loewner Evolutions. We also study the (average) integral means multifractal spectra of these unbounded whole-plane SLE curves. We prove the existence of a phase transition at a moment order p = p *(κ) > 0, at which one goes from the bulk SLE κ average integral means spectrum, as predicted by the first author (Duplantier Phys. Rev. Lett. 84:1363–1367, 2000) and established by Beliaev and Smirnov (Commun Math Phys 290:577–595, 2009) and valid for p ≤ p *(κ), to a new integral means spectrum for p ≥ p *(κ), as conjectured in part by Loutsenko (J Phys A Math Gen 45(26):265001, 2012). The latter spectrum is, furthermore, shown to be intimately related, via the associated packing spectrum, to the radial SLE derivative exponents obtained by Lawler, Schramm and Werner (Acta Math 187(2):237–273, 2001), and to the local SLE tip multifractal exponents obtained from quantum gravity by the first author (Duplantier Proc. Sympos. Pure Math. 72(2):365–482, 2004). This is generalized to the integral means spectrum of the m-fold transform of the unbounded whole-plane SLE map. A succinct, preliminary, version of this study first appeared in Duplantier et al. (Coefficient estimates for whole-plane SLE processes, Hal-00609774, 2011)