Convergence of the self-avoiding walk on random quadrangulations to SLE$_{8/3}$ on $\sqrt{8/3}$-Liouville quantum gravity

Ewain Gwynne, Jason Miller

Published 2016-08-02Version 1

Let $(Q_{\mathrm{zip}} , \lambda_{\mathrm{zip}})$ be a uniform infinite quadrangulation of the half-plane decorated by a self-avoiding walk (SAW). We prove that $(Q_{\mathrm{zip}} , \lambda_{\mathrm{zip}})$ converges in the scaling limit to the metric gluing of two independent Brownian half-planes identified along their positive boundary rays. Combined with other work of the authors, this implies the convergence of the SAW on a random quadrangulation to SLE$_{8/3}$ on a certain $\sqrt{8/3}$-Liouville quantum gravity surface. The topology of convergence is the local Gromov-Hausdorff-Prokhorov-uniform topology, the natural generalization of the local Gromov-Hausdorff topology to curve-decorated metric measure spaces. We also prove analogous scaling limit results for uniform infinite quadrangulations of the whole plane decorated by either a one-sided or two-sided SAW.