TY - JOUR

T1 - A mating-of-trees approach for graph distances in random planar maps

AU - Gwynne, Ewain

AU - Holden, Nina

AU - Sun, Xin

N1 - Funding Information:
We thank an anonymous referee for helpful comments on an earlier version of this article. We thank Jason Miller for helpful discussions. E.G. was partially funded by NSF Grant DMS 1209044. N.H. was supported by a fellowship from the Norwegian Research Council. X.S. was supported by the Simons Foundation as a Junior Fellow at Simons Society of Fellows.
Publisher Copyright:
© 2020, The Author(s).

PY - 2020/8/1

Y1 - 2020/8/1

N2 - We introduce a general technique for proving estimates for certain random planar maps which belong to the γ-Liouville quantum gravity (LQG) universality class for γ∈ (0 , 2). The family of random planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d. increments via a mating-of-trees bijection, and includes the uniform infinite planar triangulation (UIPT; γ=8/3); and planar maps weighted by the number of different spanning trees (γ=2), bipolar orientations (γ=4/3), or Schnyder woods (γ= 1) that can be put on the map. Using our technique, we prove estimates for graph distances in the above family of random planar maps. In particular, we obtain non-trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the Watabiki (Prog Theor Phys Suppl 114:1–17, 1993) prediction for the Hausdorff dimension of γ-LQG and we establish the existence of an exponent for certain distances in the map. The basic idea of our approach is to compare a given random planar map M to a mated-CRT map—a random planar map constructed from a correlated two-dimensional Brownian motion—using a strong coupling (Zaitsev in ESAIM Probab Stat 2:41–108, 1998) of the encoding walk for M and the Brownian motion used to construct the mated-CRT map. This allows us to deduce estimates for graph distances in M from the estimates for graph distances in the mated-CRT map which we proved (using continuum theory) in a previous work. In the special case when γ=8/3, we instead deduce estimates for the 8/3-mated-CRT map from known results for the UIPT. The arguments of this paper do not directly use SLE/LQG, and can be read without any knowledge of these objects.

AB - We introduce a general technique for proving estimates for certain random planar maps which belong to the γ-Liouville quantum gravity (LQG) universality class for γ∈ (0 , 2). The family of random planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d. increments via a mating-of-trees bijection, and includes the uniform infinite planar triangulation (UIPT; γ=8/3); and planar maps weighted by the number of different spanning trees (γ=2), bipolar orientations (γ=4/3), or Schnyder woods (γ= 1) that can be put on the map. Using our technique, we prove estimates for graph distances in the above family of random planar maps. In particular, we obtain non-trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the Watabiki (Prog Theor Phys Suppl 114:1–17, 1993) prediction for the Hausdorff dimension of γ-LQG and we establish the existence of an exponent for certain distances in the map. The basic idea of our approach is to compare a given random planar map M to a mated-CRT map—a random planar map constructed from a correlated two-dimensional Brownian motion—using a strong coupling (Zaitsev in ESAIM Probab Stat 2:41–108, 1998) of the encoding walk for M and the Brownian motion used to construct the mated-CRT map. This allows us to deduce estimates for graph distances in M from the estimates for graph distances in the mated-CRT map which we proved (using continuum theory) in a previous work. In the special case when γ=8/3, we instead deduce estimates for the 8/3-mated-CRT map from known results for the UIPT. The arguments of this paper do not directly use SLE/LQG, and can be read without any knowledge of these objects.

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U2 - 10.1007/s00440-020-00969-8

DO - 10.1007/s00440-020-00969-8

M3 - Article

AN - SCOPUS:85082870792

SN - 0178-8051

VL - 177

SP - 1043

EP - 1102

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 3-4

ER -