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

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.