Conformal embedding and percolation on the uniform triangulation

Seminar | April 17 | 3-4 p.m. | 1011 Evans Hall

 Xin Sun, Columbia University

 Department of Statistics

Following Smirnov’s proof of Cardy’s formula and Schramm’s discovery of SLE, a thorough understanding of the scaling limit of critical percolation on the regular triangular lattice has been achieved. Smirnorv’s proof in fact gives a discrete approximation of the conformal embedding which we call the Cardy embedding. In this talk I will present a joint project with Nina Holden where we show that the uniform triangulation under the Cardy embedding converges to the Brownian disk under the conformal embedding. Moreover, we prove a quenched scaling limit result for the critical percolation on uniform triangulations. Time permitting, I will also explain how this result fits into the the larger picture of random planar maps and Liouville quantum gravity.