## Probability Seminar: Bridge Decomposition of Restriction Measures

- Date: 01/27/2010

University of British Columbia

In the early 60s Kesten showed that self-avoiding walk in the upper

half plane has a decomposition into an i.i.d. sequence of "irreducible

bridges". Loosely defined, a bridge is a self-avoiding path that

achieves its minimum and maximum heights at the start and end of the

path (respectively), and it is irreducible if it contains no smaller

bridges. Considering only the 2-dimensional case, one can ask if the

(likely) scaling limit of self-avoiding walk, the SLE(8/3) process,

also has such a decomposition. I will talk about

recent work with Hugo Duminil from Ecole Normale Superieure that

provides a positive answer, using only the restriction property of

SLE(8/3). In the end we are able to decompose the SLE(8/3) path as a

Poisson Point Process on the space of irreducible bridges, in a way

that is similar to Ito's excursion decomposition of a Brownian motion

according to its zeros. Our decomposition can actually be generalized

beyond SLE(8/3) and applied to an entire family of "restriction

measures", hence the title of the talk. If time permits I will also

talk about the natural time parameterization for SLE(8/3), which has

immediate applications towards the bridge decomposition.

4:00pm - 5:00pm, WMAX 216