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 c M ∈ (−∞, 1]. Via the DDK ansatz, LQG can equivalently be described as the random geometry obtained by exponentiating γ times a variant of the planar Gaussian free field, where γ ∈ (0, 2] satisfies c M = 25 − 6(2/γ + γ/2) 2 . Physics considerations suggest that LQG should also make sense in the regime when c M > 1. However, the behavior in this regime is rather mysterious in part because the corresponding value of 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 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 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 c M ∈ (1, 25), which gives a finite quantum dimension if and only if the Euclidean dimension is at most (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 c M ∈ (1, 25) is infinite).We include a substantial list of open problems.

Weak LQG metrics and Liouville first passage percolation

“…It is still an open problem to establish uniqueness of the scaling limit for LGD. Similar considerations apply to variants of LGD defined using embedded planar maps (such as maps constructed from LQG square subdivision [16,19] or mated-CRT maps [21,27]) instead of Euclidean balls, although for these variants tightness has not been checked.…”

“…In particular, with Q(ξ ) as above, the central charge should be related to ξ by c = 25 − 6Q(ξ ) 2 . See [3,13,14,19,30] for further discussion of this extended phase of LQG and some justification for the above predictions.…”

Punctures and p-spin curves from matrix models III. Dl type and logarithmic potential

“…Note that we used λ 2 for HZ transform instead of λ, thus above result becomes same as (9.25). For p = −3 with HZ transform, one obtains behavior like a branched polymer, which is related to quantum Liouville field [45]. The area of 1 < C for p < 0 is interesting in the relation to gauge theory as we discussed in chapter 9.…”

Up-to-constants comparison of Liouville first passage percolation and Liouville quantum gravity

“…The case when c M ∈ (1, 25) is much less understood, even from a physics perspective. See [GHPR20,DG20] for further discussion.…”