Renormalized Brownian loop soup layering fields converge in an appropriate Sobolev sense to a tilted imaginary Gaussian multiplicative chaos with covariance kernel given by the Brownian loop measure.
Integrability of Liouville theory: proof of the DOZZ Formula
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abstract
Dorn and Otto (1994) and independently Zamolodchikov and Zamolodchikov (1996) proposed a remarkable explicit expression, the so-called DOZZ formula, for the 3 point structure constants of Liouville Conformal Field Theory (LCFT), which is expected to describe the scaling limit of large planar maps properly embedded into the Riemann sphere. In this paper we give a proof of the DOZZ formula based on a rigorous probabilistic construction of LCFT in terms of Gaussian Multiplicative Chaos given earlier by F. David and the authors. This result is a fundamental step in the path to prove integrability of LCFT, i.e. to mathematically justify the methods of Conformal Bootstrap used by physicists. From the purely probabilistic point of view, our proof constitutes the first rigorous integrability result on Gaussian Multiplicative Chaos measures.
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Brownian Loops, Layering Fields and Imaginary Gaussian Multiplicative Chaos
Renormalized Brownian loop soup layering fields converge in an appropriate Sobolev sense to a tilted imaginary Gaussian multiplicative chaos with covariance kernel given by the Brownian loop measure.