Random walk on random planar maps: Spectral dimension, resistance and displacement

Abstract

We study simple random walk on the class of random planar maps which can be encoded by a two-dimensional random walk with i.i.d.. increments or a two-dimensional Brownian motion via a “mating-of-trees” type bijection. This class includes the uniform infinite planar triangulation (UIPT), the infinite-volume limits of random planar maps weighted by the number of spanning trees, bipolar orientations, or Schnyder woods they admit, and the γ-mated-CRT map for $\mathit{\gamma }\in (0,2)$. For each of these maps, we obtain an upper bound for the Green’s function on the diagonal, an upper bound for the effective resistance to the boundary of a metric ball, an upper bound for the return probability of the random walk to its starting point after n steps, and a lower bound for the graph-distance displacement of the random walk, all of which are sharp up to polylogarithmic factors.

When combined with work of Lee (2017), our bound for the return probability shows that the spectral dimension of each of these random planar maps is a.s. equal to 2, that is, the (quenched) probability that the simple random walk returns to its starting point after $2\mathit{n}$ steps is ${\mathit{n}}^{-1+{\mathit{o}}_{\mathit{n}}(1)}$. Our results also show that the amount of time that it takes a random walk to exit a metric ball is at least its volume (up to a polylogarithmic factor). In the special case of the UIPT, this implies that random walk typically travels at least ${\mathit{n}}^{1/4-{\mathit{o}}_{\mathit{n}}(1)}$ units of graph distance in n units of time. The matching upper bound for the displacement is proven by Gwynne and Hutchcroft (Probab. Theory Related Fields 178 (2020) 567–611). These two works together resolve a conjecture of Benjamini and Curien (Geom. Funct. Anal. 23 (2013) 501–531) in the UIPT case.

Our proofs are based on estimates for the mated-CRT map (which come from its relationship to SLE-decorated Liouville quantum gravity) and a strong coupling of the mated-CRT map with the other random planar map models.