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Generalized integral means spectrum of whole-plane SLE Zinsmeister, Michel


(Joint work with B.Duplantier, H.Ho and B.Le).\\ If $f$ is a holomorphic and injective map from the unit disk into the complex plane normalized by $f(0)=0,\,f'(0)=1$ one defines its "integral means spectrum" as $p\mapsto \beta_f(p)$ where $\beta_f(p)$ is such that $$\int_0^{2\pi}\vert f'(re^{it})\vert^p dt\sim (1-r)^{-\beta_f(p)}$$ as $r\to 1$. The universal integral means spectrum is the supremum of the set of $\{\beta_f(p)\}$, $f$ over the set of all normalized conformal maps. It is well-known that if we restrict to bounded conformal maps one obtains a different spectrum. The first aim of this talk is to define a generalized integral means spectrum as $(p,q)\mapsto \beta_f(p,q)$ where $$\int_0^{2\pi}\frac{\vert f'(re^{it})\vert^p}{\vert f(re^{it})\vert^q} dt\sim (1-r)^{-\beta_f(p,q)}$$ allowing to unify the bounded/unbounded approach.\\ We put to use this new concept to study the generalized integral means spectrum of the whole-plane SLE; the case $q=2p$ corresponds to the external (bounded) case studied by Beliaev and Smirnov. There is a gap the latter paper: the proof fails for $p<p^*=-\frac{(4+\kappa)^2(8+\kappa)}{128}$. Using the generalized spectrum we show that this value $p^*$ is not an artefact of the method, but correspond to a (hidden) phase transition towards a new spectrum.

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