Samuel Watson (Brown)

Relating a classical planar map embedding algorithm to Liouville quantum gravity and SLE(16)

Abstract: In 1990, Walter Schnyder introduced a way to endow a simple planar triangulation with awood—a triple of spanning trees—which gives rise to a combinatorially natural grid embedding of the triangulation. It turns out that a uniformly sampled wooded triangulation on n vertices converges in the large-n limit to a random fractal surface (called Liouville quantum gravity) together with a triple of intertwined fractal curves (called SLE(16)). We will motivate this result by describing Schnyder’s algorithm and discussing some history of random planar map convergence results, and we will also explain the role of LQG and SLE in the story.