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Let $\rho$ be compactly supported on $D \subset \mathbb R^2$. Endow $\mathbb R^2$ with the metric $e^{\rho}(dx_1^2 + dx_2^2)$. As $\delta \to 0$ the set of Brownian loops centered in $D$ with length at least $\delta$ has measure $$\frac{\text{area}(D)}{2\pi \delta} + \frac{1}{48\pi}(\rho,\rho)_{\nabla}+ o(1).$$ When $\rho$ is smooth, this follows from the classical Polyakov-Alvarez formula. We show that the above also holds if $\rho$ is not smooth, e.g. if $\rho$ is only Lipschitz. This fact can alternatively be expressed in terms of heat kernel traces, eigenvalue asymptotics, or zeta regularized determinants. Variants of this statement apply to more general non-smooth manifolds on which one considers all loops (not only those centered in a domain $D$). We also show that the $o(1)$ error is uniform for any family of $\rho$ satisfying certain conditions. This implies that if we weight a measure $\nu$ on this family by the ($\delta$-truncated) Brownian loop soup partition function, and take the vague $\delta \to 0$ limit, we obtain a measure whose Radon-Nikodym derivative with respect to $\nu$ is $\exp\bigl( \frac{1}{48\pi}(\rho,\rho)_{\nabla}\bigr)$. When the measure is a certain regularized Liouville quantum gravity measure, a companion work [APPS20] shows that this weighting has the effect of changing the so-called central charge of the surface.

Although lattice Yang-Mills theory on finite subgraphs of $\mathbb Z^d$ is easy to rigorously define, the construction of a satisfactory continuum theory on $\mathbb R^d$ is a major open problem when $d \geq 3$. Such a theory should in some sense assign a Wilson loop expectation to each suitable finite collection $\mathcal L$ of loops in $\mathbb R^d$. One classical approach is to try to represent this expectation as a sum over surfaces with boundary $\mathcal L$. There are some formal/heuristic ways to make sense of this notion, but they typically yield an ill-defined difference of infinities. In this paper, we show how to make sense of Yang-Mills integrals as surface sums for $d=2$, where the continuum theory is more accessible. Applications include several new explicit calculations, a new combinatorial interpretation of the master field, and a new probabilistic proof of the Makeenko-Migdal equation.

We explore the geometric meaning of the so-called zeta-regularized determinant of the Laplace-Beltrami operator on a compact surface, with or without boundary. We relate the $(-c/2)$-th power of the determinant of the Laplacian to the appropriately regularized partition function of a Brownian loop soup of intensity $c$ on the surface. This means that, in a certain sense, decorating a random surface by a Brownian loop soup of intensity $c$ corresponds to weighting the law of the surface by the $(-c/2)$-th power of the determinant of the Laplacian. Next, we introduce a method of regularizing a Liouville quantum gravity (LQG) surface (with some matter central charge parameter $\mathbf{c}$) to produce a smooth surface. And we show that weighting the law of this random surface by the $( -\mathbf{c}'/ 2)$-th power of the Laplacian determinant has precisely the effect of changing the matter central charge from $\mathbf{c}$ to $\mathbf{c} + \mathbf{c}'$. Taken together with the earlier results, this provides a way of interpreting an LQG surface of matter central charge $\mathbf{c}$ as a pure LQG surface decorated by a Brownian loop soup of intensity $\mathbf{c}$. Building on this idea, we present several open problems about random planar maps and their continuum analogs. Although the original construction of LQG is well-defined only for $\mathbf{c}\leq 1$, some of the constructions and questions also make sense when $\mathbf{c}>1$., 37 pages, 5 figures; minor changes

Physics considerations suggest that a theory of Liouville quantum gravity (LQG) should exist for all values of matter central charge $\mathbf{c} \in (-\infty,25)$. Probabilists have rigorously defined LQG as a random metric measure space for $\mathbf{c} < 1$; however, they have only very recently begun to explore LQG in the $\mathbf{c} \in (1,25)$ phase. We define a random metric associated to LQG for all $\mathbf{c} < 25$ by a collection of axioms analogous to axioms stated in the $\mathbf{c} < 1$ setting. We show that such a metric exists for each $\mathbf{c}$ by considering an approximating metric known as Liouville first passage percolation. Ding and Gwynne proved that these approximating metrics are tight in a suitably chosen topology; we show that every subsequential limit satisfies our axioms. In particular, our result defines a metric associated to LQG in the critical case $\mathbf{c} =1$. The metrics we define for $\mathbf{c} \in (1,25)$ exhibit geometric behavior that sharply contrasts with the $\mathbf{c} < 1$ regime. We show that, for $\mathbf{c} \in (1,25)$, the metrics do not induce the Euclidean topology since they a.s. have a dense (measure zero) set of singular points, points at infinite distance from all other points. We use this fact to prove that a.s. the metric ball is not compact and its boundary has infinite Hausdorff dimension. Despite these differences, we demonstrate that many properties of LQG metrics for $\mathbf{c} < 1$ extend in some form to the entire range $(-\infty, 25)$. We show that the metrics are a.s. reverse H��lder continuous with respect to the Euclidean metric, are a.s. complete and geodesic away from the set of singular points, and satisfy the bounds for set-to-set distances that hold in the $\mathbf{c} < 1$ phase. Finally, we prove that the metrics satisfy a version of the (geometric) Knizhnik-Polyakov-Zamolodchikov (KPZ) formula., 56 pages, 4 figures

Let $M$ be the infinite spanning-tree-weighted random planar map, which is the local limit of finite random planar maps sampled with probability proportional to the number of spanning trees they admit. We show that a.s. the $M$-graph-distance diameter of the external diffusion-limited aggregation (DLA) cluster on $M$ run for $m$ steps is of order $m^{2/d + o_m(1)}$, where $d$ is the metric ball volume growth exponent for $M$ (which was shown to exist by Ding-Gwynne, 2018). By known bounds for $d$, one has $0.55051\ldots \leq 2/d \leq 0.563315\ldots$. Along the way, we also prove that loop-erased random walk (LERW) on $M$ typically travels graph distance $m^{2/d + o_m(1)}$ in $m$ units of time and that the graph-distance diameter of a finite spanning-tree-weighted random planar map with $n$ edges, with or without boundary, is of order $n^{1/d+o_n(1)}$ except on an event with probability decaying faster than any negative power of $n$. Our proofs are based on a special relationship between DLA and LERW on spanning-tree-weighted random planar maps as well as estimates for distances in such maps which come from the theory of Liouville quantum gravity., Comment: 49 pages, 8 figures; to appear in AOP

We study the adjacency graph of bubbles---i.e., complementary connected components---of an SLE$_��$ curve for $��\in (4,8)$, with two such bubbles considered to be adjacent if their boundaries intersect. We show that this adjacency graph is a.s. connected for $��\in (4,��_0]$, where $��_0 \approx 5.6158$ is defined explicitly. This gives a partial answer to a problem posed by Duplantier, Miller and Sheffield (2014). Our proof in fact yields a stronger connectivity result for $��\in (4,��_0]$, which says that there is a Markovian way of finding a path from any fixed bubble to $\infty$. We also show that there is a (non-explicit) $��_1 \in (��_0, 8)$ such that this stronger condition does not hold for $��\in [��_1,8)$. Our proofs are based on an encoding of SLE$_��$ in terms of a pair of independent $��/4$-stable processes, which allows us to reduce our problem to a problem about stable processes. In fact, due to this encoding, our results can be re-phrased as statements about the connectivity of the adjacency graph of loops when one glues together an independent pair of so-called $��/4$-stable looptrees, as studied, e.g., by Curien and Kortchemski (2014). The above encoding comes from the theory of Liouville quantum gravity (LQG), but the paper can be read without any knowledge of LQG if one takes the encoding as a black box., 28 pages, 5 figures; final version, to appear in AOP

We show that random walk on a stationary random graph with positive anchored expansion and exponential volume growth has positive speed. We also show that two families of random triangulations of the hyperbolic plane, the hyperbolic Poisson Voronoi tessellation and the hyperbolic Poisson Delaunay triangulation, have 1-skeletons with positive anchored expansion. As a consequence, we show that the simple random walks on these graphs have positive speed. We include a section of open problems and conjectures on the topics of stationary geometric random graphs and the hyperbolic Poisson Voronoi tessellation., Comment: 33 pages, 2 figures. Updated to include corrections to Proposition 1.5 and Lemma 4.4