**Extremal conformal metrics, spectral geometry, and distributional limits of graphs**

I will talk about an intrinsic approach to uniformization of a graph's geometry based on extremal discrete metrics. This method allows one to generalize known results about planar graphs that rely heavily on the theory of circle packings, and to obtain new information even for well-studied models of planar graphs like the uniform infinite triangulation (UIPT) and quadrangulation (UIPQ).

It yields a short proof of Benjamini and Schramm's result that a distributional limit of bounded-degree planar graphs is almost surely recurrent. The same argument resolves a 2001 conjecture of those authors since it works also for H-minor-free graphs (and, in fact, a substantial generalization known as region intersection graphs).

Gurel-Gurevich and Nachmias recently solved a central open problem by showing that UIPT and UIPQ are almost surely recurrent. By combining extremal discrete metrics with methods from spectral geometry, I will present a new proof of this fact that also gives explicit control on divergence of the Green function. If \(g(x, T)\) denotes the number of returns to the root at time T, we show that almost surely (over the choice of the random rooted graph), \(g(x,T)\) grows asymptotically at least as fast as \(\log \log T\). This is a consequence of a general result that holds for any distributional limit of H-minor-free graphs and provides lower bounds on return probabilities based on the tails of the degree distribution of the root.

Finally, I will discuss extensions to general graphs and, in particular, give a characterization of the almost sure spectral dimension of a distributional limit in terms of the "conformal growth rate" of any sequence that approaches the limit graph. This has consequences for graphs that can be sphere-packed in \(\mathbb{R}^d\) for \(d > 2\).