Columbia Probability Seminar

The seminar covers a wide range of topics in pure and applied probability. The seminar is organized jointly by the Mathematics and Statistics departments and is run by Julien Dubedat, Clement Hongler, Fredrik Johansson Viklund, Ioannis Karatzas, and Philip Protter.

E-mail the organizers at probability_seminar -- at -- math.columbia.edu.

In the spring, the seminar usually takes place in Statistics SSW Building Room 903 (1255 Amsterdam Avenue between 121st & 122nd Street) on Fridays at 12:00noon-1:00pm.

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Last seminar for the Spring Semester: Friday April 22: Gabor Pete (University of Toronto)

The scaling limits of dynamical and near-critical percolation, and the Minimal Spanning Tree

Let each site of the triangular lattice, with small mesh $\eta$, have an independent Poisson clock with a certain rate $r(\eta) = \eta^{3/4+o(1)}$ switching between open and closed such that, at any given moment, the configuration is just critical percolation. In particular, the probability of a left-right open crossing in the unit square is close to 1/2. Furthermore, because of the scaling, the expected number of switches in unit time between having a crossing or not is of unit order.

We prove that the limit (as $\eta \to 0$) of the above process exists as a Markov process, and it is conformally covariant: if we change the domain with a conformal map $\phi(z)$, then time has to be scaled locally by $|\phi'(z)|^{3/4}$. A key step in the proof is that the counting measure on the set of pivotal points in a percolation configuration has a conformally covariant scaling limit. Similar limit measures can be constructed for other special points: length measures on interfaces and area measures on clusters.

The same proof yields a similar result for near-critical percolation, and it also shows that the scaling limit of (a version of) the Minimal Spanning Tree exists, it is invariant under translations, rotations and scaling, but *probably* not under general conformal maps. I will explain what is missing for a proof.

Joint works with Christophe Garban and Oded Schramm.