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

We prove that a uniform infinite quadrangulation of the half-plane decorated by self-avoiding walk (SAW) converges in scaling limit to metric gluing two independent Brownian half-planes identified along their positive boundary rays. Combined with other work authors, this implies convergence SAW on random SLE$_{8/3}$ certain $\sqrt{8/3}$-Liouville quantum gravity surface. The topology is local Gromov-Hausdorff-Prokhorov-uniform topology, natural generalization Gromov-Hausdorff curve-decorated measure spaces. also analogous results for quadrangulations whole plane either one-sided or two-sided SAW. Our proof uses only peeling procedure and some basic properties half-plane, so can be read without any knowledge SLE LQG.