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Mardi 6 mars 2012-14:00

Exact probability of percolation hulls touching the boundary

Yacine IKHLEF (Université de Genève)

par Bertrand Georgeot - 6 mars 2012

The main objects of study in 2D percolation are the percolation clusters and the lattice curves surrounding them (hulls). In the scaling limit, Conformal Field Theory (CFT) and SLE allow the determination of correlation functions of these hulls. It turns out that some of these functions can be computed in finite size, opening a way to rigorous proofs of conformal invariance and universality, and uncovering some less understood aspects of CFT. I will present how, using transfer-matrix methods (Yang-Baxter, qKZ), we obtained an exact expression of the probability for a percolation hull to touch the boundary, on a strip of finite width L. We also relate the left-passage probability in the Fortuin-Kasteleyn cluster model to the magnetisation profile in the open XXZ chain : this is a new geometrical application of Bethe-Ansatz techniques developped for spin chains