A mating-of-trees approach for graph distances in random planar maps
Ewain Gwynne,Nina Holden,Xin Sun +2 more
TLDR
In this paper, the authors introduce a general technique for proving estimates for certain random planar maps which belong to the Liouville quantum gravity (LQG) universality class.Abstract:
We introduce a general technique for proving estimates for certain random planar maps which belong to the
$$\gamma $$
-Liouville quantum gravity (LQG) universality class for
$$\gamma \in (0,2)$$
. 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;
$$\gamma =\sqrt{8/3}$$
); and planar maps weighted by the number of different spanning trees (
$$\gamma =\sqrt{2}$$
), bipolar orientations (
$$\gamma =\sqrt{4/3}$$
), or Schnyder woods (
$$\gamma =1$$
) 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
$$\gamma $$
-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
$$\gamma =\sqrt{8/3}$$
, we instead deduce estimates for the
$$\sqrt{8/3}$$
-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.read more
Citations
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Mating of trees for random planar maps and Liouville quantum gravity: a survey
Ewain Gwynne,Nina Holden,Xin Sun +2 more
TL;DR: The mating-of-trees theorem of Duplantier, Miller, and Sheffield as mentioned in this paper gives an encoding of a Liouville quantum gravity surface decorated by a Schramm-Loewner evolution (SLE) curve in terms of a pair of correlated linear Brownian motions.
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Existence and uniqueness of the Liouville quantum gravity metric for $\gamma \in (0,2)$
Ewain Gwynne,Jason Miller +1 more
TL;DR: In this article, it was shown that the subsequential limit of the Liouville first passage percolation (LFPP) admits non-trivial subsequential limits.
Journal ArticleDOI
The Fractal Dimension of Liouville Quantum Gravity: Universality, Monotonicity, and Bounds
Jian Ding,Ewain Gwynne +1 more
TL;DR: In this paper, it was shown that for any planar map in the LQG universality class, there is an exponent for the Liouville heat kernel and exponents for various continuum approximations of planar distances.
Journal ArticleDOI
A distance exponent for Liouville quantum gravity
Ewain Gwynne,Nina Holden,Xin Sun +2 more
TL;DR: In this article, the LQG structure graphs (a.k.a. mated-CRT maps) were studied and upper and lower bounds for the cardinality of a graph-distance ball of radius n in the Gromov-Hausdorff topology were derived.
References
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Gregory F. Lawler,Vlada Limic +1 more
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Embedding planar graphs on the grid
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