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Title:
Multifractality of Whole-Plane SLE
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Abstract:
We revisit the Bieberbach conjecture in the framework of the SLE process. The study of its unbounded whole-plane version 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 $m$-fold conformal maps of whole-plane SLE. We 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 $p=p^*(\kappa)>0$, at which one goes from the bulk SLE$_\kappa$ average integral means spectrum, valid for $p\leq p^*(\kappa)$, to a new integral means spectrum for $p\geq p^*(\kappa)$. The latter spectrum is furthermore shown to be intimately related, via the associated packing spectrum, to the so-called radial SLE derivative exponents, and to the non-standard, local SLE tip multifractal exponents obtained from quantum gravity. This is generalized to the integral means spectrum of the $m$-fold transform of the unbounded whole-plane SLE map. Joint work with Nguyen T.P. Chi, Nguyen T.T. Nga and Michel Zinsmeister
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