Probability Seminar: Bridge Decomposition of Restriction Measures

  • Date: 01/27/2010
Tom Alberts (University of Toronto)

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