Convergence of uniform triangulations under the Cardy embedding
Nina Holden, Xin Sun
Abstract
Definition 1.1.(Cardy embedding) Given a triangulation of a polygon M with three distinct boundary edges a, b , and c ordered counterclockwise, let Ber M be the probability measure corresponding to the Bernoulli-1 2 site percolation on M .The Cardy embedding Cdy M of (M, a, b, c) is the function from V(M ) to ∆ given bySmirnov's theorem [Sm] can be phrased in terms of the Cardy embedding as follows.Suppose that D is a Jordan domain with three distinct marked boundary points a, b, and c ordered counterclockwise.Let T denote the triangular lattice.Given a small mesh uniform triangulations under the cardy embedding n. holden and x. sun Now to conclude the proof of (1.4), we just need to show that as ε!0, the ε-cutoff dynamic of the discrete LDP driven by e h/ √ 6 stabilize to a limiting ergodic process.