SLE and the free field: partition functions and couplings

By Julien Dubédat. 2007

247 citations recorded by the source corpus.

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 $\zeta$-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 $\kappa>0$.

Subfields: Computational Theory and Mathematics, Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/092cbca57e2055a1270da3895f7daebf43ed7fa3

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Citations in the corpus

Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees (2013, background): “This step is carried out in some generality in [3, 26, 14].”
Conformal loop ensembles: the Markovian characterization and the loop-soup construction (2010, background): “The outermost boundaries defined by the union of two independent CLE3s in a domain (recall [7] that CLE3 is the scaling limit of the Ising model loops, and note that it corresponds to c = 1/2) form a CLE4 (which corresponds to “outermost” level lines of the Gaussian Free Field; see [35], [10] and to”
Liouville Quantum Gravity on the Riemann Sphere (2014, methodology): “This gives rise to the notion of Gaussian Free Field Xĝ with vanishing mean in the metric of the sphere ĝ, which is a random distribution living almost surely in the dual Sobolev space H−1(R2, ĝ) (see [18]).”
The energy of a deterministic Loewner chain: Reversibility and interpretation via SLE$_{0+}$ (2016, background): “The quotient term of Poisson excursion kernel does not depend on the local coordinates (for more discussions on the Poisson excursion kernel, readers may refer to [6] Section 3).”
Imaginary geometry I: interacting SLEs (2012, methodology): “of a result in [Dub09b], though the generalization is non-trivial and one of the major new contributions of this article.”
Imaginary geometry II: Reversibility of SLEκ(ρ1;ρ2) for κ∈(0,4). (2016, result): “This step is carried out for certain boundary conditions in [Dub09b] and in more generality in [MS12a].”
A contour line of the continuum Gaussian free field (2010, background): “3 appears in [Dub09], which references our work in progress and also explores relationships between this coupling and continuum partition functions, Laplacian determinants, and the Polyakov-Alvarez conformal anomaly formula.”
Conformal weldings of random surfaces: SLE and the quantum gravity zipper (2010, methodology): “1, it was shown in [Dub09] that in any such coupling, ηT is almost surely equal to a particular path-valued function of h.”