An SLE2 loop measure

作者：Benoist Stephane^*; Dubedat Julien

来源：Annales de l Institut Henri Poincare-Probabilites et Statistiques, 2016, 52(3): 1406-1436.

DOI：10.1214/15-AIHP681

• SLE
• UST
• Loop
• Restriction

There is an essentially unique way to associate to any Riemann surface a measure on its simple loops, such that the collection of measures satisfy a strong conformal invariance property (see (J. Amer. Math. Soc. 21 (2008) 137-169)). These random loops are constructed as the boundary of Brownian loops, and so correspond in the zoo of statistical mechanics models to central charge 0, or Schramm-Loewner Evolution (SLE) parameter kappa = 8/3. The goal of this paper is to construct a family of measures on simple loops on Riemann surfaces that satisfies a conformal covariance property, and that would correspond to SLE parameter kappa = 2 (central charge -2). On planar annuli, this loop measure was already built by Adrien Kassel and Rick Kenyon in (Random curves on surfaces induced from the Laplacian determinant (2012) ArXiv e-prints). We will give an alternative construction of this loop measure on planar annuli, investigate its conformal covariance, and finally extend this measure to general Riemann surfaces. This gives an example of a Malliavin-Kontsevich-Suhov loop measure (Tr. Mat. Inst. Steklova 258 (2007) 107-153) in non-zero central charge.

• 出版日期2016-8