Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach
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 describe...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Springer Berlin Heidelberg
2021
|
| Online Access: | https://hdl.handle.net/1721.1/131446 |
| _version_ | 1826213804393889792 |
|---|---|
| author | Gwynne, Ewain Holden, Nina Pfeffer, Joshua Remy, Guillaume |
| author_facet | Gwynne, Ewain Holden, Nina Pfeffer, Joshua Remy, Guillaume |
| author_sort | Gwynne, Ewain |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T15:55:05Z |
| format | Article |
| id | mit-1721.1/131446 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T15:55:05Z |
| publishDate | 2021 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1314462021-09-21T03:08:41Z Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach Gwynne, Ewain Holden, Nina Pfeffer, Joshua Remy, Guillaume 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. 2021-09-20T17:17:07Z 2021-09-20T17:17:07Z 2020-01-16 2020-09-24T20:51:56Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/131446 en https://doi.org/10.1007/s00220-019-03663-6 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 Pfeffer, Joshua Remy, Guillaume Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach |
| title | Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach |
| title_full | Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach |
| title_fullStr | Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach |
| title_full_unstemmed | Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach |
| title_short | Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach |
| title_sort | liouville quantum gravity with matter central charge in 1 25 a probabilistic approach |
| url | https://hdl.handle.net/1721.1/131446 |
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