A distance exponent for Liouville quantum gravity

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 )

.