Area measures and branched polymers in supercritical Liouville quantum gravity

TL;DR

This paper demonstrates that supercritical Liouville quantum gravity surfaces, when conditioned to be finite, converge to the continuum random tree, linking physics predictions with rigorous mathematical results.

Contribution

It proves that supercritical LQG conditioned on finiteness converges to the continuum random tree, and shows the non-existence of a locally finite measure satisfying LQG coordinate change formulas.

Findings

Supercritical LQG conditioned on finiteness converges to the continuum random tree.

No locally finite measure exists for supercritical LQG satisfying the LQG coordinate change formula.

The proofs use a branching process description derived from coupling with CLE$_4$.

Abstract

We study Liouville quantum gravity (LQG) in the supercritical (a.k.a. strongly coupled) phase, which has background charge $Q \in (0,2)$ and central charge $\mathbf{c}_{\mathrm{L}} = 1+6Q^2 \in (1,25)$. Recent works have shown how to define LQG in this phase as a planar random geometry associated with a variant of the Gaussian free field, which exhibits "infinite spikes." In contrast, a number of results from physics, dating back to the 1980s, suggest that supercritical LQG surfaces should behave like "branched polymers": i.e., they should look like the continuum random tree. We prove a result which reconciles these two descriptions of supercritical LQG. More precisely, we show that for a family of random planar maps with boundary in the universality class of supercritical LQG, if we condition on the (small probability) event that the planar map is finite, then the scaling limit is the continuum random tree. We also show that there does not exist any locally finite measure associated with supercritical LQG which is locally determined by the field and satisfies the LQG coordinate change formula. Our proofs are based on a branching process description of supercritical LQG which comes from its coupling with CLE$_4$ (Ang and Gwynne, arXiv:2308.11832).