A distance exponent for Liouville quantum gravity
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Author(s)
Gwynne, Ewain
Holden, Nina
Sun, Xin
Date Issued
April 26, 2018
Publisher
Springer Berlin Heidelberg
Version
Author's final manuscript
Abstract
Abstract
Let
$$\gamma \in (0,2)$$
γ
∈
(
0
,
2
)
and let h be the random distribution on
$$\mathbb C$$
C
which describes a
$$\gamma $$
γ
-Liouville quantum gravity (LQG) cone. Also let
$$\kappa = 16/\gamma ^2 >4$$
κ
=
16
/
γ
2
>
4
and let
$$\eta $$
η
be a whole-plane space-filling SLE
$$_\kappa $$
κ
curve sampled independent from h and parametrized by
$$\gamma $$
γ
-quantum mass with respect to h. We study a family
$$\{\mathcal G^\epsilon \}_{\epsilon >0}$$
{
G
ϵ
}
ϵ
>
0
of planar maps associated with
$$(h, \eta )$$
(
h
,
η
)
called the LQG structure graphs (a.k.a. mated-CRT maps) which we conjecture converge in probability in the scaling limit with respect to the Gromov–Hausdorff topology to a random metric space associated with
$$\gamma $$
γ
-LQG. In particular,
$$\mathcal G^\epsilon $$
G
ϵ
is the graph whose vertex set is
$$\epsilon \mathbb Z$$
ϵ
Z
, with two such vertices
$$x_1,x_2\in \epsilon \mathbb Z$$
x
1
,
x
2
∈
ϵ
Z
connected by an edge if and only if the corresponding curve segments
$$\eta ([x_1-\epsilon , x_1])$$
η
(
[
x
1
-
ϵ
,
x
1
]
)
and
$$\eta ([x_2-\epsilon ,x_2])$$
η
(
[
x
2
-
ϵ
,
x
2
]
)
share a non-trivial boundary arc. Due to the peanosphere description of SLE-decorated LQG due to Duplantier et al. (Liouville quantum gravity as a mating of trees, 2014), the graph
$$\mathcal G^\epsilon $$
G
ϵ
can equivalently be expressed as an explicit functional of a correlated two-dimensional Brownian motion, so can be studied without any reference to SLE or LQG. We prove non-trivial upper and lower bounds for the cardinality of a graph-distance ball of radius n in
$$\mathcal G^\epsilon $$
G
ϵ
which are consistent with the prediction of Watabiki (Prog Theor Phys Suppl 114:1–17, 1993) for the Hausdorff dimension of LQG. Using subadditivity arguments, we also prove that there is an exponent
$$\chi > 0$$
χ
>
0
for which the expected graph distance between generic points in the subgraph of
$$\mathcal G^\epsilon $$
G
ϵ
corresponding to the segment
$$\eta ([0,1])$$
η
(
[
0
,
1
]
)
is of order
$$\epsilon ^{-\chi + o_\epsilon (1)}$$
ϵ
-
χ
+
o
ϵ
(
1
)
, and this distance is extremely unlikely to be larger than
$$\epsilon ^{-\chi + o_\epsilon (1)}$$
ϵ
-
χ
+
o
ϵ
(
1
)
.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00440-018-0846-9
