On the threshold for triangulations inside convex polygons

Start with a large convex polygon and add all other edges inside independently with probability p. At what critical threshold pc do triangulations of the polygon begin to appear? The first author and Gravner asked this question, and observed that pc=Θ(1), using the relationship with the Catalan numbers and a coupling with oriented site percolation on Z². More recently, Archer, Hartarsky, the first author, Olesker-Taylor, Schapira and Valesin proved that 1/4<pc<pcᵒ, where $1/4$ is the Catalan exponential growth rate and pcᵒ is the critical threshold for oriented percolation. The upper bound is strict, but non-quantitative, and follows by a renormalization argument. We show that pc<1/2 using a simple ear clipping algorithm, which can be analyzed using the gambler's ruin problem. This bound is closer to the truth (perhaps near $0.4$) and shows that most configurations of edges inside large convex polygons contain triangulations.