Abstract
A mated-CRT map is a random planar map obtained as a discretized mating of correlated continuum random trees. Mated-CRT maps provide a coarse-grained approximation of many other natural random planar map models (e.g., uniform triangulations and spanning tree-weighted maps), and are closely related to $\gamma $-Liouville quantum gravity (LQG) for $\gamma \in (0,2)$ if we take the correlation to be $-\cos (\pi \gamma ^{2}/4)$. We prove estimates for the Dirichlet energy and the modulus of continuity of a large class of discrete harmonic functions on mated-CRT maps, which provide a general toolbox for the study of the quantitative properties of random walk and discrete conformal embeddings for these maps. For example, our results give an independent proof that the simple random walk on the mated-CRT map is recurrent, and a polynomial upper bound for the maximum length of the edges of the mated-CRT map under a version of the Tutte embedding. Our results are also used in other work by the first two authors which shows that for a class of random planar maps — including mated-CRT maps and the UIPT — the spectral dimension is two (i.e., the return probability of the simple random walk to its starting point after $n$ steps is $n^{-1+o_{n}(1)}$) and the typical exit time of the walk from a graph-distance ball is bounded below by the volume of the ball, up to a polylogarithmic factor.
Highlights
There has been substantial interest in random planar maps in recent years. One reason for this is that random planar maps are the discrete analogs of γ-Liouville quantum gravity (LQG) surfaces for γ ∈ (0, 2)
We will focus on this last type of question for a particular family of random planar maps called mated-CRT maps, which are directly connected to many other random planar map models and to LQG
Our results provide a general toolbox for the study of random walk on mated-CRT maps, and thereby random walk on other random planar maps thanks to the coupling results discussed above
Summary
There has been substantial interest in random planar maps in recent years. One reason for this is that random planar maps are the discrete analogs of γ-Liouville quantum gravity (LQG) surfaces for γ ∈ (0, 2). It follows from [DMS14, Theorem 1.9] that if we let Gε for ε > 0 be the graph whose vertex set is εZ, with two vertices x1, x2 ∈ εZ connected by an edge if and only if the corresponding cells η([x1 − ε, x1]) and η([x2 − ε, x2]) share a non-trivial boundary arc, {Gε}ε>0 has the same law as the family of mated-CRT maps defined above. For many quantities of interest, one can get stronger estimates using this embedding than using circle packing since we have good estimates for the behavior of space-filling SLE and the γ-LQG measure
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