Bridge decomposition of restriction measures. (English) Zbl 1197.82055
Summary: Motivated by Kesten’s bridge decomposition for two-dimensional self-avoiding walks in the upper half plane, we show that the conjectured scaling limit of the half-plane SAW, the \(\text{SLE}(8/3)\) process, also has an appropriately defined bridge decomposition. This continuum decomposition turns out to entirely be a consequence of the restriction property of \(\text{SLE}(8/3)\), and as a result can be generalized to the wider class of restriction measures. Specifically, we show that the restriction hulls with index less than one can be decomposed into a Poisson point process of irreducible bridges in a way that is similar to Itô’s excursion decomposition of a Brownian motion according to its zeros.
MSC:
| 82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |
| 60J65 | Brownian motion |
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
Keywords:
self-avoiding walk; conformal invariance; SLE; restriction measures; Poisson process; excursion theoryReferences:
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