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

Abstract

Karl Lowner (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, Levy 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 Levy–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 SLEκ 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).