Electronic Journal of Probability

Harmonic functions on mated-CRT maps

Ewain Gwynne, Jason Miller, and Scott Sheffield

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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.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 58, 55 pp.

Received: 28 July 2018
Accepted: 23 May 2019
First available in Project Euclid: 21 June 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks 60K37: Processes in random environments 60J67: Stochastic (Schramm-)Loewner evolution (SLE)

random planar maps random walk in random environment mated-CRT map Liouville quantum gravity Schramm-Loewner evolution mating of trees

Creative Commons Attribution 4.0 International License.


Gwynne, Ewain; Miller, Jason; Sheffield, Scott. Harmonic functions on mated-CRT maps. Electron. J. Probab. 24 (2019), paper no. 58, 55 pp. doi:10.1214/19-EJP325. https://projecteuclid.org/euclid.ejp/1561082667

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