Harmonic functions on mated-CRT maps
Name
1807.07511.pdf
Description
Accepted version
Size
1.39 MB
Format
Adobe PDF
Checksum (MD5)
9bfd5ff5ce2f89c78a7762ce85e41636
Author(s)
Gwynne, Ewain
Miller, Jason Eric
Sheffield, Scott Roger
Date Issued
May 2019
Journal
Electronic journal of probability
Publisher
Institute of Mathematical Statistics
Citation
Gwynne, Ewain, Jason Miller and Scott Sheffield. “Harmonic functions on mated-CRT maps.” Electronic journal of probability, 24, 58 (May 2019) © 2019 The Author(s)
Version
Author's final manuscript
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 γ -Liouville quantum gravity (LQG) for γ ∈ (0, 2) if we take the correlation to be -cos(pγ2/4). 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+on(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.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1214/19-EJP325
