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We study simple random walk on the class of random planar maps which can beencoded by a two-dimensional random walk with i.i.d. increments or atwo-dimensional Brownian motion via a "mating-of-trees" type bijection. Thisclass includes the uniform infinite planar triangulation (UIPT), theinfinite-volume limits of random planar maps weighted by the number of spanningtrees, bipolar orientations, or Schnyder woods they admit, and the$\gamma$-mated-CRT map for $\gamma \in (0,2)$. For each of these maps, weobtain an upper bound for the Green's function on the diagonal, an upper boundfor the effective resistance to the boundary of a metric ball, an upper boundfor 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 randomwalk, all of which are sharp up to polylogarithmic factors.  When combined with work of Lee (2017), our bound for the return probabilityshows that the spectral dimension of each of these random planar maps is a.s.equal to 2, i.e., the (quenched) probability that the simple random walkreturns to its starting point after $2n$ steps is $n^{-1+o_n(1)}$. Our resultsalso show that the amount of time that it takes a random walk to exit a metricball is at least its volume (up to a polylogarithmic factor). In the specialcase of the UIPT, this implies that random walk typically travels at least$n^{1/4 - o_n(1)}$ units of graph distance in $n$ units of time. The matchingupper bound for the displacement is proven by Gwynne and Hutchcroft (2018).These two works together resolve a conjecture of Benjamini and Curien (2013) inthe UIPT case.  Our proofs are based on estimates for the mated-CRT map (which come from itsrelationship to SLE-decorated Liouville quantum gravity) and a strong couplingof the mated-CRT map with the other random planar map models.

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