2009/04/29 by Julien Dubédat · 1 citation
Mathematics · Computer Science · Physics and Astronomy · #Stochastic processes and statistical mechanics #Advanced Mathematical Modeling in Engineering #Theoretical and Computational Physics #Algorithm #Mathematics #Uniqueness #Regularization (linguistics) #Artificial intelligence #Computer science #Mathematical analysis
paper · pdf · doi:10.1090/s0894-0347-09-00636-5
openalex publication_date 2009/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/20
Schramm-Loewner Evolutions (SLE) are random curves in planar simply connected domains; the massless (Euclidean) free field in such a domain is a random distribution. Both have conformal invariance properties in law. In the present article, some relations between the two objects are studied. We establish identities of partition functions between different versions of SLE and the free field with appropriate boundary conditions; this involves <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="zeta"> <mml:semantics> <mml:mi> ζ </mml:mi> <mml:annotation encoding="application/x-tex">ζ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -regularization and the Polyakov-Alvarez conformal anomaly formula. We proceed with a construction of couplings of SLE with the free field, showing that, in a precise sense, chordal SLE is the solution of a stochastic “differential” equation driven by the free field. Existence, uniqueness in law, and pathwise uniqueness for these SDEs are proved for general <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi> κ </mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">κ >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This identifies SLE curves as local observables of the free field.