Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach

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.