Random walk on random planar maps: spectral dimension, resistance, and displacement
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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
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, i.e., the (quenched) probability that the simple random walk returns to its starting point after
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.
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2168-894X