Random surfaces and lattice Yang-Mills

We study Wilson loop expectations in lattice Yang-Mills models with a compact Lie group $G$. Using tools recently introduced in a companion paper, we provide alternate derivations, interpretations, and generalizations of several recent theorems about Brownian motion limits (Dahlqvist), lattice string trajectories (Chatterjee and Jafarov) and surface sums (Magee and Puder). We show further that one can express Wilson loop expectations as sums over embedded planar maps in a manner that applies to any matrix dimension $N \geq 1$, any inverse temperature $\beta>0$, and any lattice dimension $d \geq 2$. When $G=\mathrm{U}(N)$, the embedded maps we consider are pairs $(\mathcal M, \phi)$ where $\mathcal M$ is a planar (or higher genus) map and $\phi$ is a graph homomorphism from $\mathcal M$ to a lattice such as $\mathbb Z^d$. The faces of $\mathcal M$ come in two partite classes: $\textit{edge-faces}$ (each mapped by $\phi$ onto a single edge) and $\textit{plaquette-faces}$ (each mapped by $\phi$ onto a single plaquette). The weight of a lattice edge $e$ is the Weingarten function applied to the partition whose parts are given by half the boundary lengths of the faces in $\phi^{-1}(e)$. (The Weingarten function becomes quite simple in the $N\to \infty$ limit.) The overall weight of an embedded map is proportional to $N^\chi$ (where $\chi$ is the Euler characteristic) times the product of the edge weights. We establish analogous results for $\mathrm{SU}(N)$, $\mathrm{O}(N)$, $\mathrm{SO}(N)$, and $\mathrm{Sp}(N/2)$, where the embedded surfaces and weights take a different form. There are several variants of these constructions. In this context, we present a list of relevant open problems spanning several disciplines: random matrix theory, representation theory, statistical physics, and the theory of random surfaces, including random planar maps and Liouville quantum gravity.