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 ζ \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 κ > 0 \kappa >0 . This identifies SLE curves as local observables of the free field.

Schramm-Loewner Evolutions (SLEs) describe a one-parameter family of growth processes in the plane that have particular conformal invariance properties. For instance, SLE can define simple random curves in a simply connected domain. In this paper we are interested in questions pertaining to the definition of several SLEs in a domain (i.e. several random curves). In particular, one derives infinitesimal commutation conditions, discuss some elementary solutions, study integrability conditions following from commutation and show how to lift these infinitesimal relations to global relations in simple cases. The situation in multiply-connected domains is also discussed.For plane critical models of statistical physics, such as percolation or the Ising model, the general line of thinking of Conformal Field Theory leads to expect the existence of a non-degenerate scaling limit that satisfies conformal invariance properties. Though, it is not quite clear how to define this scaling limit and what conformal invariance exactly means.One way to proceed is to consider a model in a, say, bounded (plane) simply-connected domain with Jordan boundary, and to set boundary conditions so as to force the existence of a macroscopic interface connecting two marked points on the boundary. In this set-up, Schramm has shown that the possible scaling limits satifying conformal invariance along with a "domain Markov" property are classified by a single positive parameter κ > 0, in the seminal article [19]. This defines the family of Schramm-Loewner Evolutions (SLEs), that are probability measures supported on non-self-traversing curves connecting two marked boundary points in a simply-connected domain.Consider the following situation for critical site percolation on the triangular lattice: a portion of the triangular lattice with mesh ε approximates a fixed simply connected domain D with two points x and y marked on the boundary. The boundary arc (xy) is set to blue and (yx) is set to yellow; sites are blue or yellow with probability 1/2. Then the interface between blue sites connected to (xy) and yellow sites connected to (yx) is a non-self traversing curve from x to y. In this set-up, Smirnov has proved that the interface converges to SLE 6 , as conjectured earlier by Schramm ([21, 4]).For discrete models such as percolation or the Ising model, the full information can be encoded as a collection of contours (interfaces between blue any yellow, + and − spins, . . . ). Hence it is quite natural to consider scaling limits as collection of countours, as in [1,4]. Comparing the ideas of isolating one macroscopic interface by setting appropriate boundary conditions (following Schramm), and considering the scaling limit as a collection of contours, one is led to the problem of describing the joint law of a finite number of macroscopic interfaces created by appropriate boundary conditions. For instance, for percolation, consider a simply connected domain with 2n marked points on the boundary, the 2n boundary arcs being alternatively blue ...

Schramm-Loewner Evolutions (SLEs) have proved an efficient way to describe a single continuous random conformally invariant interface in a simply-connected planar domain; the admissible probability distributions are parameterized by a single positive parameter κ. As shown in [8], the coexistence of n interfaces in such a domain implies algebraic ("commutation") conditions. In the most interesting situations, the admissible laws on systems of n interfaces are parameterized by κ and the solution of a particular (finite rank) holonomic system.The study of solutions of differential systems, in particular their global behaviour, often involves the use of integral representations. In the present article, we provide Euler integral representations for solutions of holonomic systems arising from SLE commutation. Applications to critical percolation (general crossing formulae), Loop-Erased Random Walks (direct derivation of Fomin's formulae in the scaling limit), and Uniform Spanning Trees are discussed. The connection with conformal restriction and Poissonized non-intersection for chordal SLEs is also studied.

Ann. Scient. Éc. Norm. Sup. 4 e série, t. 42, 2009, p. 697 à 724 DUALITY OF SCHRAMM-LOEWNER EVOLUTIONS  J DUBÉDAT A. -In this note, we prove a version of the conjectured duality for Schramm-Loewner Evolutions, by establishing exact identities in distribution between some boundary arcs of chordal SLEκ, κ > 4, and appropriate versions of SLEκ,κ = 16/κ.R. -On démontre dans cette note une version de la dualité conjecturée pour les évolutions de Schramm-Loewner, en établissant des identités en distribution exactes entre certains arcs de SLEκ chordal, κ > 4, et des versions appropriées de SLEκ,κ = 16/κ.

It is known that Schramm-Loewner Evolutions (SLEs) have a.s. frontier points if κ > 4 and a.s. cutpoints if 4 < κ < 8. If κ > 4, an appropriate version of SLE(κ) has a renewal property: it starts afresh after visiting its frontier. Thus one can give an excursion decomposition for this particular SLE(κ) "away from its frontier". For 4 < κ < 8, there is a two-sided analogue of this situation: a particular version of SLE(κ) has a renewal property w.r.t its cutpoints; one studies excursion decompositions of this SLE "away from its cutpoints". For κ = 6, this overlaps Virág's results on "Brownian beads". As a by-product of this construction, one proves Watts' formula, which describes the probability of a double crossing in a rectangle for critical plane percolation.Schramm-Loewner Evolutions (SLEs) are a family of growth processes in simply connected plane domains. Their conformal invariance properties make them natural candidates to describe the scaling limit of critical plane systems, that are generally conjectured to converge to a conformally invariant limit. This convergence has been rigorously established in several cases: for instance, the scaling limit of the Loop-Erased Random Walk (resp. the Peano curve of the Uniform Spanning Tree) is SLE(2) (resp. SLE(8) ) (see [18]), and the scaling limit of critical percolation interfaces is SLE(6) for site percolation on the triangular lattice (see [26]).The qualitative features of SLE depend crucially on the value of the κ parameter. The growth process is generated by a continuous path, the trace (see [24]). This path is a.s. simple if κ ≤ 4; if κ > 4, it is no longer the case, and SLE has a non-trivial frontier ([24]). Furthermore, SLE has cutpoints if 4 < κ < 8 (see [1]).In [28], Virág shows that the Brownian Excursion (Brownian motion in the upper half-plane, started from 0 and conditioned not to hit the real line again) can be decomposed in "beads", i.e. portions of the Brownian excursion between two successive cutpoints. This decomposition can be phrased in terms similar to Itô's theory of Brownian excursions.For SLE, one also has a Markov property and conformal invariance, so it is quite natural to look for similar decompositions w.r.t. loci with an intrinsic geometrical definition. We will see that such decompositions exist (for suitably conditioned SLEs) for frontier points and cutpoints. J. DubédatWhile considering restriction formulas in [10], in relation with duality conjectures, it appeared that a particular version of SLE(κ), namely SLE(κ, κ−4), played a special role. For κ > 4, SLE(κ) a.s. swallows any real point; and SLE (0,0 + ) (κ, κ−4) can be viewed as an SLE(κ) "conditioned" not to hit the positive half-line. As this event has zero probability, a little care (and precision) is required. This process, run until infinity, has a right-boundary that is a simple path connecting 0 and ∞ in H; this right-boundary is conjectured to be identical in law to an SLE(κ , κ /2 − 2), where κκ = 16.Looking at the right-boundary as a path, it is quite natural to cons...