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A conformally invariant metric on CLE(4)

Date: 
Tuesday 27th January 2015 - 13:30 to 14:30
Venue: 
INI Seminar Room 1
Abstract: 
Co-authors: Scott Sheffield (MIT), Hao Wu (MIT)

Werner and Wu introduced a conformally invariant way of exploring the loops in a CLE$_4$ $\Gamma$ in a simply connected domain. Using the relationship between CLE$_4$ and the Gaussian free field, we show that the dynamics of this exploration process are a deterministic function of the CLE$_4$ loops, and we use this fact to construct a conformally invariant metric on $\Gamma$ for which a ball of radius $t$ coincides with the set of loops explored up to time $t$ by the exploration process. It is conjectured that this metric space is related in the $\epsilon \to 0$ limit to the contact graph metric on CLE$_{4+\epsilon}$ as well as the contact graph metric on $\epsilon$-neighborhoods of CLE$_4$ loops.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons