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Probability and Combinatorics

Tuesday, February 13, 2018 - 3:00pm

Ewain Gwynne



University of Pennsylvania


I will discuss a general strategy for proving estimates for a certain class of random planar maps, namely, those which can be encoded by a two-dimensional walk with i.i.d. increments via a ``mating-of-trees" type bijection. This class includes the uniform infinite planar triangulation (UIPT) and the infinite-volume limits of random planar maps sampled with probability proportional to the number of spanning trees, bipolar orientations, or Schnyder woods they admit.Using this strategy, we obtain non-trivial estimates for graph distances in certain natural non-uniform random planar maps. We also prove that random walk on the UIPT typically travels graph distance $n^{1/4 + o_n(1)}$ in $n$ units of time and that the spectral dimension of a class of random planar maps (including the UIPT) is a.s. equal to 2---i.e., the return probability to the starting point after $n$ steps is $n^{-1+o(1)}$.Our approach proceeds by way of a strong coupling of the encoding walk for the map with a correlated two-dimensional Brownian motion (Zaitsev, 1998), which allows us to compare our given map with the so-called mated-CRT map constructed from this correlated two-dimensional Brownian. The mated-CRT map is closely related to SLE-decorated Liouville quantum gravity due to results of Duplantier, Miller, and Sheffield (2014). So, we can analyze the mated-CRT map using continuum theory and then transfer to other random planar maps via strong coupling. We expect that this approach will have further applications in the future.Based on various joint works with Nina Holden, Tom Hutchcroft, Jason Miller, and Xin Sun.