Department of Mathematics
UChicago logo

James Lee to give Department Colloquium on March 29

Discrete conformal metrics and spectral geometry on distributional limits


Random triangulations have been studied as a discrete model for 2D quantum gravity since the 1980s.  While there are many conjectures about the emergent geometry of such models, mathematical progress has been somewhat slower.  Toward this end, Benjamini and Schramm (2001) defined the distributional limit of a sequence of finite graphs; the limit object is a unimodular random graph in the sense of Aldous and Lyons. Benjamini and Schramm showed that every distributional limit of finite planar graphs with uniformly bounded degrees is almost surely recurrent (the random walk returns to its starting point infinitely often almost surely).

Their approach uses the Koebe-Andreev-Thurston circle packing theorem to uniformize the geometry of the limit graph.  In contrast, we consider intrinsic deformations of the path metric by a (random) weighting of the vertices.  This leads to the notion of the conformal growth exponent of a unimodular random graph, which is the best degree of volume growth of balls that can be achieved by such a weighting of "unit area." The conformal growth exponent carries information about the underlying geometry; in particular, it bounds the almost sure spectral dimension.

We show that distributional limits of finite graphs that can be sphere-packed in \(\mathbb{R}^d\) have conformal growth exponent at most \(d\), and thus the connection to spectral dimension yields \(d\)-dimensional lower bounds on the heat kernel.  When the conformal growth exponent is bounded by 2, one obtains more precise information, including a conjectured generalization of the Benjamini-Schramm Recurrence Theorem to larger families of graphs.

These methods extend to models with unbounded degrees, giving new proofs of almost sure recurrence for the extensively studied uniform infinite planar triangulation (UIPT) and quadrangulation (UIPQ).  The latter results were established only recently by Gurel-Gurevich and Nachmias (2013).  Our approach yields quantitative lower bounds on the heat kernel, spectral measure, and speed of the random walk.