Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach
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Author(s)
Gwynne, Ewain
Holden, Nina
Pfeffer, Joshua
Remy, Guillaume
Date Issued
January 2020
Journal
Communications in Mathematical Physics
Publisher
Springer Berlin Heidelberg
Version
Author's final manuscript
Abstract
Abstract
There is a substantial literature concerning Liouville quantum gravity (LQG) in two dimensions with conformal matter field of central charge
$${{\mathbf {c}}}_{\mathrm M} \in (-\infty ,1]$$
c
M
∈
(
-
∞
,
1
]
. Via the DDK ansatz, LQG can equivalently be described as the random geometry obtained by exponentiating
$$\gamma $$
γ
times a variant of the planar Gaussian free field, where
$$\gamma \in (0,2]$$
γ
∈
(
0
,
2
]
satisfies
$${\mathbf {c}}_{\mathrm M} = 25 - 6(2/\gamma + \gamma /2)^2$$
c
M
=
25
-
6
(
2
/
γ
+
γ
/
2
)
2
. Physics considerations suggest that LQG should also make sense in the regime when
$${\mathbf {c}}_{\mathrm M} > 1$$
c
M
>
1
. However, the behavior in this regime is rather mysterious in part because the corresponding value of
$$\gamma $$
γ
is complex, so analytic continuations of various formulas give complex answers which are difficult to interpret in a probabilistic setting. We introduce and study a discretization of LQG which makes sense for all values of
$${\mathbf {c}}_{\mathrm M} \in (-\infty ,25)$$
c
M
∈
(
-
∞
,
25
)
. Our discretization consists of a random planar map, defined as the adjacency graph of a tiling of the plane by dyadic squares which all have approximately the same “LQG size" with respect to the Gaussian free field. We prove that several formulas for dimension-related quantities are still valid for
$$\mathbf{c}_{\mathrm M} \in (1,25)$$
c
M
∈
(
1
,
25
)
, with the caveat that the dimension is infinite when the formulas give a complex answer. In particular, we prove an extension of the (geometric) KPZ formula for
$$\mathbf{c}_{\mathrm M} \in (1,25)$$
c
M
∈
(
1
,
25
)
, which gives a finite quantum dimension if and only if the Euclidean dimension is at most
$$(25-\mathbf{c}_{\mathrm M} )/12$$
(
25
-
c
M
)
/
12
. We also show that the graph distance between typical points with respect to our discrete model grows polynomially whereas the cardinality of a graph distance ball of radius r grows faster than any power of r (which suggests that the Hausdorff dimension of LQG in the case when
$${\mathbf {c}}_{\mathrm M} \in (1,25)$$
c
M
∈
(
1
,
25
)
is infinite). We include a substantial list of open problems.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
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.
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s00220-019-03663-6
