2005/07/31 by Julien Dubédat, Julien Dubedat · 82 citations
Mathematics · Physics and Astronomy · #Algebraic number #Combinatorics #Conformal map #Euler's formula #Holonomic #Invariant (physics) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Parameterized complexity #Pure mathematics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:60G17 #msc:60G52 #msc:60K35 #msc:82B43
paper · pdf · doi:10.1007/s10955-006-9132-9
published in Journal of Statistical Physics 123(6), 1183-1218 (Springer Science+Business Media) · v3: minor revision. 28 pages. to appear in J. Statist. Phys
arxiv created 2006/05/21 · openalex publication_date 2006/06/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 \citeD6, 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.