A distance exponent for Liouville quantum gravity
Abstract Let $$\gamma \in (0,2)$$ γ ∈ (...
| Main Authors: | , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer Berlin Heidelberg
2021
|
| Online Access: | https://hdl.handle.net/1721.1/131438 |
| _version_ | 1826210896476635136 |
|---|---|
| author | Gwynne, Ewain Holden, Nina Sun, Xin |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Gwynne, Ewain Holden, Nina Sun, Xin |
| author_sort | Gwynne, Ewain |
| collection | MIT |
| description | 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
)
. |
| first_indexed | 2024-09-23T14:57:30Z |
| format | Article |
| id | mit-1721.1/131438 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T14:57:30Z |
| publishDate | 2021 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1314382024-01-02T20:01:29Z A distance exponent for Liouville quantum gravity Gwynne, Ewain Holden, Nina Sun, Xin Massachusetts Institute of Technology. Department of Mathematics 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 ) . 2021-09-20T17:17:05Z 2021-09-20T17:17:05Z 2018-04-26 2020-09-24T20:57:54Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/131438 en https://doi.org/10.1007/s00440-018-0846-9 Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer-Verlag GmbH Germany, part of Springer Nature application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Gwynne, Ewain Holden, Nina Sun, Xin A distance exponent for Liouville quantum gravity |
| title | A distance exponent for Liouville quantum gravity |
| title_full | A distance exponent for Liouville quantum gravity |
| title_fullStr | A distance exponent for Liouville quantum gravity |
| title_full_unstemmed | A distance exponent for Liouville quantum gravity |
| title_short | A distance exponent for Liouville quantum gravity |
| title_sort | distance exponent for liouville quantum gravity |
| url | https://hdl.handle.net/1721.1/131438 |
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