Euler Integrals for Commuting SLEs

By Julien Dubédat. 2005

95 citations recorded by the source corpus.

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, Ref. 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.

Subfields: Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9379b0369faf2e76ff76bb0a3c47de35d7eca3b0

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Citations in the corpus

The configurational measure on mutually avoiding SLE paths (2006, methodology): “The formula in the following proposition has appeared previously; see, for example, [3].”
SLE and the free field: partition functions and couplings (2007, background): “This is analyzed combinatorially in [18]; see also [9, 7].”
Commutation relations for Schramm‐Loewner evolutions (2004, methodology): “In fact, as is discussed in [10], one can make sense of this sytem for any κ ∈ (0, 8/3) using appropriate loop-soups.”