91社区

Title: Area measures and branched polymers in supercritical Liouville quantum gravity.

Abstract: Liouville quantum gravity (LQG) is a "canonical" one-parameter model of surfaces with random geometry, where the parameter c >1 is the central charge of the associated conformal field theory. Compared to the subcritical and critical phases with c 鈮� 25 (corresponding to 饾浘 鈮� 2), much less is known about the geometry of LQG in the supercritical phase c 鈭� (1,25). Recent work of Ding and Gwynne has shown how to construct LQG in this phase as a planar random geometry associated with 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 look like the continuum random tree.

In this talk, I will give a result that reconciles these two descriptions. More precisely, for a family of random planar maps 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. Separately, we 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. Both results are based on a branching process description of supercritical LQG which comes from its coupling with CLE_4 by Ang and Gwynne.

This is joint work with Manan Bhatia and Ewain Gwynne.

听