Search

Fractional Gaussian fields are scalar-valued random functions or generalized functions on an $n$-dimensional manifold $M$, indexed by a parameter $s$. They include white noise ($s = 0$), Brownian motion ($s=1, n=1$), the 2D Gaussian free field ($s = 1, n=2$) and the membrane model ($s = 2$). These simple objects are ubiquitous in math and science, and can be used as a starting point for constructing non-Gaussian theories. The $\textit{differential form}$ analogs of these objects are equally natural: for example, instead of considering an instance $h(x)$ of the GFF on $\mathbb R^2$, one might write $h_1(x)dx_1 + h_2(x) dx_2$ where $h_1$ and $h_2$ are independent GFF instances. In general, given $k \in \{0,1,\ldots,n\}$, an instance of the $\textit{fractional Gaussian $k$-form}$ with parameter $s \in \mathbb R$ (abbreviated $\mathrm{FGF}_s^k(M)$) is given by $(-\Delta)^{-\frac{s}{2}} W_k,$ where $W_k$ is a $k$-form-valued white noise. We write $$\textrm{FGF}_s^k(M)_{d=0} \quad \textrm{and} \quad \textrm{FGF}_s^k(M)_{d^*=0}$$ for the $L^2$ orthogonal projections of $\textrm{FGF}_s^k(M)$ onto the space of $k$-forms on which $d$ (resp.\ $d^*$) vanishes. We explain how $\mathrm{FGF}_s^k(M)$ and its projections transform under $d$ and $d^*$, as well as wedge/Hodge-star operators, subspace restrictions, and axial projections. We discuss how the $1$-form $\textrm{FGF}_1^1(M)$ and its $\textit{gauge-fixed}$ projection $\textrm{FGF}_1^1(M)_{d^*=0}$ are related to gauge theories, and we formulate several conjectures and open problems about scaling limits, including possible off-critical/non-Gaussian limits, whose construction in the Yang-Mills setting is a famous open problem., Comment: 84 pages, 17 figures

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., Comment: Adjusted exposition in many places to improve clarity. 127 pages, 59 figures

Associated to two given sequences of eigenvalues $\lambda_1 \geq \dots \geq \lambda_n$ and $\mu_1 \geq \dots \geq \mu_n$ is a natural polytope, the polytope of augmented hives with the specified boundary data, which is associated to sums of random Hermitian matrices with these eigenvalues. As a first step towards the asymptotic analysis of random hives, we show that if the eigenvalues are drawn from the GUE ensemble, then the associated augmented hives exhibit concentration as $n \rightarrow \infty$. Our main ingredients include a representation due to Speyer of augmented hives involving a supremum of linear functions applied to a product of Gelfand--Tsetlin polytopes; known results by Klartag on the KLS conjecture in order to handle the aforementioned supremum; covariance bounds of Cipolloni--Erd\H{o}s--Schr\"oder of eigenvalue gaps of GUE; and the use of the theory of determinantal processes to analyze the GUE minor process., Comment: 45 pages, 15 figures

The Smith embedding of a finite planar map with two marked vertices, possibly with conductances on the edges, is a way of representing the map as a tiling of a finite cylinder by rectangles. In this embedding, each edge of the planar map corresponds to a rectangle, and each vertex corresponds to a horizontal segment. Given a sequence of finite planar maps embedded in an infinite cylinder, such that the random walk on both the map and its planar dual converges to Brownian motion modulo time change, we prove that the a priori embedding is close to an affine transformation of the Smith embedding at large scales. By applying this result, we prove that the Smith embeddings of mated-CRT maps with the sphere topology converge to $\gamma$-Liouville quantum gravity ($\gamma$-LQG)., Comment: 53 pages, 12 figures. Accepted for publication in the Annals of Probability

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., Comment: Appendix A has been added for the companion paper "Random surfaces and lattice Yang-Mills" by S. Cao, M. Park, and S. Sheffield

In 2000, Cohn, Kenyon and Propp studied uniformly random perfect matchings of large induced subgraphs of $\mathbb Z^2$ (a.k.a. dimer configurations or domino tilings) and developed a large deviation theory for the associated height functions. We establish similar results for large induced subgraphs of $\mathbb Z^3$. To formulate these results, recall that a perfect matching on a bipartite graph induces a flow that sends one unit of current from each even vertex to its odd partner. One can then subtract a "reference flow'' to obtain a divergence-free flow. We show that the flow induced by a uniformly random dimer configuration converges in law (when boundary conditions on a bounded $R \subset \mathbb R^3$ are controlled and the mesh size tends to zero) to the deterministic divergence-free flow $g$ on $R$ that maximizes $$\int_{R} \text{ent}(g(x)) \,dx$$ given the boundary data, where $\text{ent}(s)$ is the maximal specific entropy obtained by an ergodic Gibbs measure with mean current $s$. The function $\text{ent}$ is not known explicitly, but we prove that it is continuous and {\em strictly concave} on the octahedron $\mathcal O$ of possible mean currents (except on the edges of $\mathcal O$) which implies (under reasonable boundary conditions) that the maximizer is uniquely determined. We further establish two versions of a large deviation principle, using the integral above to quantify how exponentially unlikely the discrete random flows are to approximate other deterministic flows. The planar dimer model is mathematically rich and well-studied, but many of the most powerful tools do not seem readily adaptable to higher dimensions. Our analysis begins with a smaller set of tools, which include Hall's matching theorem, the ergodic theorem, non-intersecting-lattice-path formulations, and double-dimer cycle swaps., Comment: Minor revisions, added index of symbols and terms. 156 pages, 40 figures

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.

Given $2n$ unit equilateral triangles, there are finitely many ways to glue each edge to a partner. We obtain a random sphere-homeomorphic surface by sampling uniformly from the gluings that produce a topological sphere. As $n$ tends to infinity, these random surfaces (appropriately scaled) converge in law. The limit is a "canonical" sphere-homeomorphic random surface, much the way Brownian motion is a canonical random path. Depending on how the surface space and convergence topology are specified, the limit is the Brownian sphere, the peanosphere, the pure Liouville quantum gravity sphere, or a certain conformal field theory. All of these objects have concise definitions, and are all in some sense equivalent, but the equivalence is highly non-trivial, building on hundreds of math and physics papers over the past half century. More generally, the "continuum random surface embedded in $d$-dimensional Euclidean space" makes a kind of sense for $d \in (-\infty, 25)$ even when $d$ is not a positive integer; and this can be extended to higher genus surfaces, surfaces with boundary, and surfaces with marked points or other decoration. These constructions have deep roots in both mathematics and physics, drawing from classical graph theory, complex analysis, probability and representation theory, as well as string theory, planar statistical physics, random matrix theory and a simple model for two-dimensional quantum gravity. We present here an informal, colloquium-level overview of the subject, which we hope will be accessible to both newcomers and experts. We aim to answer, as cleanly as possible, the fundamental question. What is a random surface?, Comment: 57 pages, 32 figures. Proceedings of the ICM contribution for 2022

Suppose $\alpha, \beta$ are Lipschitz strongly concave functions from $[0, 1]$ to $\mathbb{R}$ and $\gamma$ is a concave function from $[0, 1]$ to $\mathbb{R}$, such that $\alpha(0) = \gamma(0) = 0$, and $\alpha(1) = \beta(0) = 0$ and $\beta(1) = \gamma(1) = 0.$ For an $n \times n$ Hermitian matrix $W$, let $spec(W)$ denote the vector in $\mathbb{R}^n$ whose coordinates are the eigenvalues of $W$ listed in non-increasing order. Let $\lambda = \partial^- \alpha$, $\mu = \partial^- \beta$ on $(0, 1]$ and $\nu = \partial^- \gamma,$ at all points of $(0, 1]$, where $\partial^-$ is the left derivative, which is monotonically decreasing. Let $\lambda_n(i) := n^2(\alpha(\frac{i}{n})-\alpha(\frac{i-1}{n}))$, for $i \in [n]$, and similarly, $\mu_n(i) := n^2(\beta(\frac{i}{n})-\beta(\frac{i-1}{n}))$, and $\nu_n(i) := n^2(\gamma(\frac{i}{n})-\gamma(\frac{i-1}{n}))$. Let $X_n, Y_n$ be independent random Hermitian matrices from unitarily invariant distributions with spectra $\lambda_n$, $\mu_n$ respectively. We define norm $\|\cdot\|_\mathcal{I}$ to correspond in a certain way to the sup norm of an antiderivative. For suitable $\lambda$ and $\mu$, we prove that the following limit exists. \begin{equation} \lim\limits_{n \rightarrow \infty}\frac{\ln \mathbb{P}\left[\|spec(X_n + Y_n) - \nu_n\|_{\mathcal{I}} < n^2 \epsilon\right]}{n^2}.\end{equation} We interpret this limit in terms of the surface tension $\sigma$ of continuum limits of the discrete hives defined by Knutson and Tao.

Recent works have shown that there is a canonical way to to assign a metric (distance function) to a Liouville quantum gravity (LQG) surface for any parameter $\gamma \in (0,2)$. We establish a strong confluence property for LQG geodesics, which generalizes a result proven by Angel, Kolesnik and Miermont for the Brownian map. Using this property, we also establish zero-one laws for the Hausdorff dimensions of geodesics, metric ball boundaries, and metric nets w.r.t. the Euclidean or LQG metric. In the case of a metric ball boundary, our result combined with earlier work of Gwynne (2020) gives a formula for the a.s. Hausdorff dimension for the boundary of the metric ball stopped when it hits a fixed point in terms of the Hausdorff dimension of the whole LQG surface. We also show that the Hausdorff dimension of the metric ball boundary is carried by points which are not on the boundary of any complementary connected component of the ball., Comment: 44 pages, 9 figures, to appear in PTRF

We consider the low-prevalence linearized SEIR epidemic model for a society that has resolved to keep future infections low in anticipation of a vaccine. The society can vary its amount of potentially-infection-spreading activity over time, within a certain feasible range. Because the activity has social or economic value, the society aims to maximize activity overall subject to infection rate constraints. We find that consistent policies are the worst possible in terms of activity, while the best policies alternate between high and low activity. In a variant involving multiple subpopulations, we find that the best policies are maximally coordinated (maintaining similar prevalence among subpopulations) but oscillatory (having growth rates that vary in time). It turns out that linearized SEIR is mathematically equivalent to an idealized racecar model (with different subpopulations corresponding to different cars) and the amount of fuel used corresponds to the amount of activity. Using this analogy, steady V-shaped formations (in which one subpopulation "leads the way" with consistently higher prevalence and activity, while others follow behind with lower prevalence and activity) are especially problematic. These formations are very effective at minimizing fuel use, hence very ineffective at boosting activity. In an appendix, we obtain analogous results for alternative notions of activity, which incorporate crowding effects., Comment: 15 pages, 6 figures

We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble CLE$_{\kappa'}$ for $\kappa'$ in $(4,8)$ that is drawn on an independent $\gamma$-LQG surface for $\gamma^2=16/\kappa'$. The results are similar in flavor to the ones from our paper dealing with CLE$_{\kappa}$ for $\kappa$ in $(8/3,4)$, where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the CLE$_{\kappa'}$ in terms of stable growth-fragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on decorated planar maps. This has consequences for questions that do a priori not involve LQG surfaces: Our previous paper "CLE percolations" described the law of interfaces obtained when coloring the loops of a CLE$_{\kappa'}$ independently into two colors with respective probabilities $p$ and $1-p$. This description was complete up to one missing parameter $\rho$. The results of the present paper about CLE on LQG allow us to determine its value in terms of $p$ and $\kappa'$. It shows in particular that CLE$_{\kappa'}$ and CLE$_{16/\kappa'}$ are related via a continuum analog of the Edwards-Sokal coupling between FK$_q$ percolation and the $q$-state Potts model (which makes sense even for non-integer $q$ between $1$ and $4$) if and only if $q=4\cos^2(4\pi /\kappa')$. This provides further evidence for the long-standing belief that CLE$_{\kappa'}$ and CLE$_{16/\kappa'}$ represent the scaling limits of FK$_q$ percolation and the $q$-Potts model when $q$ and $\kappa'$ are related in this way. Another consequence of the formula for $\rho(p,\kappa')$ is the value of half-plane arm exponents for such divide-and-color models (a.k.a. fuzzy Potts models) that turn out to take a somewhat different form than the usual critical exponents for two-dimensional models., Comment: Dedicated to the memory of Harry Kesten. To appear in Probab. Theory Rel. Fields, 30 pages, 18 figures

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 $\mathbf{c}>1$., Comment: 37 pages, 5 figures; minor changes

We show that when one draws a simple conformal loop ensemble (CLE$_\kappa$ for $\kappa \in (8/3,4)$) on an independent $\sqrt{\kappa}$-Liouville quantum gravity (LQG) surface and explores the CLE in a natural Markovian way, the quantum surfaces (e.g., corresponding to the interior of the CLE loops) that are cut out form a Poisson point process of quantum disks. This construction allows us to make direct links between CLE on LQG, asymmetric $(4/\kappa)$-stable processes, and labeled branching trees. The ratio between positive and negative jump intensities of these processes turns out to be $-\cos (4 \pi / \kappa)$, which can be interpreted as a "density" of CLE loops in the CLE on LQG setting. Positive jumps correspond to the discovery of a CLE loop (where the LQG length of the loop is given by the jump size) and negative jumps correspond to the moments where the discovery process splits the remaining to be discovered domain into two pieces. Some consequences are the following: (i) It provides a construction of a CLE on LQG as a patchwork/welding of quantum disks. (ii) It allows to construct the "natural quantum measure" that lives in a CLE carpet. (iii) It enables us to derive some new properties and formulas for SLE processes and CLE themselves (without LQG) such as the exact distribution of the trunk of the general asymmetric SLE$_\kappa(\kappa-6)$ processes. The present work deals directly with structures in the continuum and makes no reference to discrete models, but our calculations match those for scaling limits of O(N) models on planar maps with large faces and CLE on LQG. Indeed, our L\'evy-tree descriptions are exactly the ones that appear in the study of the large-scale limit of peeling of discrete decorated planar maps such as in recent work of Bertoin, Budd, Curien and Kortchemski. The case of non-simple CLEs on LQG is the topic of another paper., Comment: 46 pages, final version to appear in Ann. Probab

Graph homomorphisms from the $\mathbb{Z}^d$ lattice to $\mathbb{Z}$ are functions on $\mathbb{Z}^d$ whose gradients equal one in absolute value. These functions are the height functions corresponding to proper $3$-colorings of $\mathbb{Z}^d$ and, in two dimensions, corresponding to the $6$-vertex model (square ice). We consider the uniform model, obtained by sampling uniformly such a graph homomorphism subject to boundary conditions. Our main result is that the model delocalizes in two dimensions, having no translation-invariant Gibbs measures. Additional results are obtained in higher dimensions and include the fact that every Gibbs measure which is ergodic under even translations is extremal and that these Gibbs measures are stochastically ordered., Comment: 21 pages, 2 Figures. Corrected Lemma 4.2 (Lemma 4.12 in the current version) from the previous version

Recent works have shown that an instance of a Brownian surface (such as the Brownian map or Brownian disk) a.s. has a canonical conformal structure under which it is equivalent to a $\sqrt{8/3}$-Liouville quantum gravity (LQG) surface. In particular, Brownian motion on a Brownian surface is well-defined. The construction in these works is indirect, however, and leaves open a basic question: is Brownian motion on a Brownian surface the limit of simple random walk on increasingly fine discretizations of that surface, the way Brownian motion on $\mathbb R^2$ is the $\epsilon \to 0$ limit of simple random walk on $\epsilon \mathbb Z^2$? We answer this question affirmatively by showing that Brownian motion on a Brownian surface is (up to time change) the $\lambda \to \infty$ limit of simple random walk on the Voronoi tessellation induced by a Poisson point process whose intensity is $\lambda$ times the associated area measure. Among other things, this implies that as $\lambda \to \infty$ the Tutte embedding (a.k.a. harmonic embedding) of the discretized Brownian disk converges to the canonical conformal embedding of the continuum Brownian disk, which in turn corresponds to $\sqrt{8/3}$-LQG. Along the way, we obtain other independently interesting facts about conformal embeddings of Brownian surfaces, including information about the Euclidean shapes of embedded metric balls and Voronoi cells. For example, we derive moment estimates that imply, in a certain precise sense, that these shapes are unlikely to be very long and thin., Comment: 49 pages and 2 figures

There are many classical random walk in random environment results that apply to ergodic random planar environments. We extend some of these results to random environments in which the length scale varies from place to place, so that the law of the environment is in a certain sense only translation invariant {\em modulo scaling}. For our purposes, an ``environment'' consists of an infinite random planar map embedded in $\mathbb C$, each of whose edges comes with a positive real conductance. Our main result is that under modest constraints (translation invariance modulo scaling together with the finiteness of a type of specific energy) a random walk in this kind of environment converges to Brownian motion modulo time parameterization in the quenched sense. Environments of the type considered here arise naturally in the study of random planar maps and Liouville quantum gravity. In fact, the results of this paper are used in separate works to prove that certain random planar maps (embedded in the plane via the so-called Tutte embedding) have scaling limits given by SLE-decorated Liouville quantum gravity, and also to provide a more explicit construction of Brownian motion on the Brownian map. However, the results of this paper are much more general and can be read independently of that program. One general consequence of our main result is that if a translation invariant (modulo scaling) random embedded planar map and its dual have finite energy per area, then they are close on large scales to a minimal energy embedding (the harmonic embedding). To establish Brownian motion convergence for an {\em infinite} energy embedding, it suffices to show that one can perturb it to make the energy finite., Comment: 62 pages, 10 figures

A mated-CRT map is a random planar map obtained as a discretized mating of correlated continuum random trees. Mated-CRT maps provide a coarse-grained approximation of many other natural random planar map models (e.g., uniform triangulations and spanning tree-weighted maps), and are closely related to $\gamma$-Liouville quantum gravity (LQG) for $\gamma \in (0,2)$ if we take the correlation to be $-\cos(\pi\gamma^2/4)$. We prove estimates for the Dirichlet energy and the modulus of continuity of a large class of discrete harmonic functions on mated-CRT maps, which provide a general toolbox for the study of the quantitative properties random walk and discrete conformal embeddings for these maps. For example, our results give an independent proof that the simple random walk on the mated-CRT map is recurrent, and a polynomial upper bound for the maximum length of the edges of the mated-CRT map under a version of the Tutte embedding. Our results are also used in other work by the first two authors which shows that for a class of random planar maps --- including mated-CRT maps and the UIPT --- the spectral dimension is two (i.e., the return probability of the simple random walk to its starting point after $n$ steps is $n^{-1+o_n(1)}$) and the typical exit time of the walk from a graph-distance ball is bounded below by the volume of the ball, up to a polylogarithmic factor., Comment: 56 pages, 8 figures. Many results in this paper were contained in a previous version of arXiv:1705.11161

The Schelling model, introduced by Schelling in 1969 as a model for residential segregation in cities, describes how populations of multiple types self-organize to form homogeneous clusters of one type. In this model, vertices in an $N$-dimensional lattice are initially assigned types randomly. As time evolves, the type at a vertex $v$ has a tendency to be replaced with the most common type within distance $w$ of $v$. We present the first mathematical description of the dynamical scaling limit of this model as $w$ tends to infinity and the lattice is correspondingly rescaled. We do this by deriving an integro-differential equation for the limiting Schelling dynamics and proving almost sure existence and uniqueness of the solutions when the initial conditions are described by white noise. The evolving fields are in some sense very "rough" but we are able to make rigorous sense of the evolution. In a key lemma, we show that for certain Gaussian fields $h$, the supremum of the occupation density of $h-\phi$ at zero (taken over all $1$-Lipschitz functions $\phi$) is almost surely finite, thereby extending a result of Bass and Burdzy. In the one dimensional case, we also describe the scaling limit of the limiting clusters obtained at time infinity, thereby resolving a conjecture of Brandt, Immorlica, Kamath, and Kleinberg., Comment: 49 pages, 8 figures

We prove that the Tutte embeddings (a.k.a. harmonic/barycentric embeddings) of certain random planar maps converge to $\gamma$-Liouville quantum gravity ($\gamma$-LQG). Specifically, we treat mated-CRT maps, which are discretized matings of correlated continuum random trees, and $\gamma$ ranges from $0$ to $2$ as one varies the correlation parameter. We also show that the associated space-filling path on the embedded map converges to space-filling SLE$_{\kappa}$ for $\kappa =16/\gamma^2$ (in the annealed sense) and that simple random walk on the embedded map converges to Brownian motion (in the quenched sense). This work constitutes the first proof that a discrete conformal embedding of a random planar map converges to LQG. Many more such statements have been conjectured. Since the mated-CRT map can be viewed as a coarse-grained approximation to other random planar maps (the UIPT, tree-weighted maps, bipolar-oriented maps, etc.), our results indicate a potential approach for proving that embeddings of these maps converge to LQG as well. To prove the main result, we establish several (independently interesting) theorems about LQG surfaces decorated by space-filling SLE. There is a natural way to use the SLE curve to divide the plane into "cells" corresponding to vertices of the mated-CRT map. We study the law of the shape of the origin-containing cell, in particular proving moments for the ratio of its squared diameter to its area. We also give bounds on the degree of the origin-containing cell and establish a form of ergodicity for the entire configuration. Ultimately, we use these properties to show (with the help of a general theorem proved in a separate paper) that random walk on these cells converges to a time change of Brownian motion, which in turn leads to the Tutte embedding result., Comment: 46 pages, 11 figures

We show that when observing the range of a chordal SLE$_\kappa$ curve for $\kappa \in (4,8)$, it is not possible to recover the order in which the points have been visited. We also derive related results about conformal loop ensembles (CLE): (i) The loops in a CLE$_\kappa$ for $\kappa \in (4,8)$ are not determined by the CLE$_\kappa$ gasket. (ii) The continuum percolation interfaces defined in the fractal carpets of conformal loop ensembles CLE$_{\kappa}$ for $\kappa \in (8/3, 4)$ (we defined these percolation interfaces in previous work, and showed there that they are SLE$_{16/\kappa}$ curves) are not determined by the CLE$_{\kappa}$ carpet that they are defined in., Comment: Final version. 44 pages, including an appendix on bichordal resampling. To appear in J. Europ. Math. Soc

Previous works in this series have shown that an instance of a $\sqrt{8/3}$-Liouville quantum gravity (LQG) sphere has a well-defined distance function, and that the resulting metric measure space (mm-space) agrees in law with the Brownian map (TBM). In this work, we show that given just the mm-space structure, one can a.s. recover the LQG sphere. This implies that there is a canonical way to parameterize an instance of TBM by the Euclidean sphere (up to M\"obius transformation). In other words, an instance of TBM has a canonical conformal structure. The conclusion is that TBM and the $\sqrt{8/3}$-LQG sphere are equivalent. They ultimately encode the same structure (a topological sphere with a measure, a metric, and a conformal structure) and have the same law. From this point of view, the fact that the conformal structure a.s. determines the metric and vice-versa can be understood as a property of this unified law. The results of this work also imply that the analogous facts hold for Brownian and $\sqrt{8/3}$-LQG surfaces with other topologies., Comment: 33 pages

We derive a surprising correspondence between SLE$_{\kappa}(\rho)$ processes and light cones of the Gaussian free field (GFF). Recall that (one-sided, chordal, origin-seeded) SLE$_\kappa(\rho)$ processes are in some sense the simplest and most natural variants of the Schramm-Loewner evolution. They were originally defined only for $\rho > -2$, but one can use L\'evy compensation to extend the definition to any $\rho > -2-\tfrac{\kappa}{2}$ and to obtain qualitatively different curves. The triangle $T = \{(\kappa, \rho): (-2-\tfrac{\kappa}{2})\vee (\tfrac{\kappa}{2}-4) < \rho < -2 \}$ is the primary focus of this paper. When $(\kappa, \rho) \in T$, the SLE$_\kappa(\rho)$ curves are highly non-simple (and double points are dense) even though $\kappa < 4$. Let $h$ be an instance of the GFF. Fix $\kappa\in (0,4)$ and $\chi = 2/\sqrt{\kappa} - \sqrt{\kappa}/2$. Recall that an imaginary geometry ray is a flow line of $e^{i(h/\chi +\theta)}$ that looks locally like SLE$_\kappa$. The light cone with parameter $\theta \in [0, \pi]$ is the set of points reachable from the origin by a sequence of rays with angles in $[-\theta/2, \theta/2]$. When $\theta=0$, the light cone looks like SLE$_\kappa$, and when $\theta = \pi$ it looks like the range of an SLE$_{16/\kappa}$. We find that when $\theta \in (0, \pi)$ the light cones are either fractal carpets with a dense set of holes or space-filling regions with no holes. We show that every non-space-filling light cone (with $\theta \in (0,\pi]$ and $\kappa \in (0,4)$) agrees in law with the range of an SLE$_\kappa(\rho)$ process with $(\kappa, \rho) \in T$. Conversely, the range of any SLE$_\kappa(\rho)$ with $(\kappa,\rho) \in T$ agrees in law with a non-space-filling light cone. As a consequence, we obtain the first proof that these SLE$_\kappa(\rho)$ processes are continuous and show that they are natural path-valued functions of the GFF., Comment: 38 pages, 13 figures

Square ice is a statistical mechanics model for two-dimensional ice, widely believed to have a conformally invariant scaling limit. We associate a Peano (space filling) curve to a square ice configuration, and more generally to a so-called 6-vertex model configuration, and argue that its scaling limit is a space-filling version of the random fractal curve SLE$_\kappa$, Schramm--Loewner evolution with parameter $\kappa$, where $4<\kappa\leq 12+8\sqrt{2}$. For square ice, $\kappa=12$. At the "free-fermion point" of the 6-vertex model, $\kappa=8+4\sqrt{3}$. These unusual values lie outside the classical interval $2\le \kappa\le 8$., Comment: 5 pages, 5 figures

In Liouville quantum gravity (or $2d$-Gaussian multiplicative chaos) one seeks to define a measure $\mu^h = e^{\gamma h(z)} dz$ where $h$ is an instance of the Gaussian free field on a planar domain $D$. Since $h$ is a distribution, not a function, one needs a regularization procedure to make this precise: for example, one may let $h_\epsilon(z)$ be the average value of $h$ on the circle of radius $\epsilon$ centered at $z$ (or an analogous average defined using a bump function supported inside that circle) and then write $\mu^h = \lim_{\epsilon \to 0} \epsilon^{\frac{\gamma^2}{2}} e^{\gamma h_\epsilon(z)} dz$. If $\phi: \tilde D \to D$ is a conformal map, one can write $\tilde h = h \circ \phi + Q \log |\phi'|$, where $Q = 2/\gamma + \gamma/2$. The measure $\mu^{\tilde h}$ on $\tilde D$ is then a.s.\ equivalent to the pullback via $\phi^{-1}$ of the measure $\mu^h$ on $D$. Interestingly, although this a.s.\ holds for each \textit{given} $\phi$, nobody has ever proved that it a.s.\ holds \textit {simultaneously} for all possible $\phi$. We will prove that this is indeed the case. This is conceptually important because one frequently defines a \textit{quantum surface} to be an equivalence class of pairs $(D, h)$ (where pairs such as the $(D,h)$ and $(\tilde D, \tilde h)$ above are considered equivalent) and it is useful to know that the set of pairs $(D,\mu^{h})$ obtained from the set of pairs $(D,h)$ in an equivalence class is itself an equivalence class with respect to the usual measure pullback relation., Comment: 21 pages, 3 figures. All comments are welcome

We endow the $\sqrt{8/3}$-Liouville quantum gravity sphere with a metric space structure and show that the resulting metric measure space agrees in law with the Brownian map. Recall that a Liouville quantum gravity sphere is a priori naturally parameterized by the Euclidean sphere ${\mathbf S}^2$. Previous work in this series used quantum Loewner evolution (QLE) to construct a metric $d_{\mathcal Q}$ on a countable dense subset of ${\mathbf S}^2$. Here we show that $d_{\mathcal Q}$ a.s. extends uniquely and continuously to a metric $\bar{d}_{\mathcal Q}$ on all of ${\mathbf S}^2$. Letting $d$ denote the Euclidean metric on ${\mathbf S}^2$, we show that the identity map between $({\mathbf S}^2, d)$ and $({\mathbf S}^2, \bar{d}_{\mathcal Q})$ is a.s. H\"older continuous in both directions. We establish several other properties of $({\mathbf S}^2, \bar{d}_{\mathcal Q})$, culminating in the fact that (as a random metric measure space) it agrees in law with the Brownian map. We establish analogous results for the Brownian disk and plane. Our proofs involve new estimates on the size and shape of QLE balls and related quantum surfaces, as well as a careful analysis of $({\mathbf S}^2, \bar{d}_{\mathcal Q})$ geodesics., Comment: 129 pages, 16 figures

Conformal loop ensembles are random collections of loops in a simply connected domain, whose laws are characterized by a natural conformal invariance property. The set of points not surrounded by any CLE loop is a natural random and conformally invariant analog of the Sierpinski gasket or carpet. In the present paper, we derive a direct relationship between each CLE consisting of simple disjoint loops (CLE($\kappa$) with $\kappa$ between 8/3 and 4) and the corresponding CLE($\kappa'$) where $\kappa':=16/\kappa$, a CLE consisting of non-disjoint loops. This is the continuum analog of the Edwards-Sokal coupling (between the q-state Potts model and the associated FK random cluster model) and its generalization to non-integer q. Like its discrete analog, our continuum correspondence has two directions. First, we show that one can construct (variants of) CLE($\kappa$) as follows: sample a CLE($\kappa'$), then use a biased coin to independently color each loop one of two colors, and then consider the outer boundaries of the clusters of loops of a given color. Second, we show how to interpret CLE($\kappa'$) loops as interfaces of a continuum analog of critical percolation within a CLE($\kappa)$ carpet. This is the first description of continuous percolation interfaces in fractal domains. These constructions provide new interpretations of the relationship between CLEs and the Gaussian free field. Along the way, we obtain results about generalized SLE$(\kappa;\rho)$ curves, and define a continuous family of natural CLE variants called boundary conformal loop ensembles (BCLEs) that share some (but not all) of the conformal symmetries that characterize CLEs, and that should be scaling limits of critical models with special boundary conditions. We extend the CLE correspondence to a BCLE correspondence that makes sense for all $\kappa$ between 2 and 4., Comment: 77 pages, 59 figures. Revised and expanded

We give bijections between bipolar-oriented (acyclic with unique source and sink) planar maps and certain random walks, which show that the uniformly random bipolar-oriented planar map, decorated by the "peano curve" surrounding the tree of left-most paths to the sink, converges in law with respect to the peanosphere topology to a $\sqrt{4/3}$-Liouville quantum gravity surface decorated by an independent Schramm-Loewner evolution with parameter $\kappa=12$ (i.e., SLE$_{12}$). This result is universal in the sense that it holds for bipolar-oriented triangulations, quadrangulations, $k$-angulations, and maps in which face sizes are mixed., Comment: 34 pages, 9 figures

Liouville quantum gravity (LQG) and the Brownian map (TBM) are two distinct models of measure-endowed random surfaces. LQG is defined in terms of a real parameter $\gamma$, and it has long been believed that when $\gamma = \sqrt{8/3}$, the LQG sphere should be equivalent (in some sense) to TBM. However, the LQG sphere comes equipped with a conformal structure, and TBM comes equipped with a metric space structure, and endowing either one with the other's structure has been an open problem for some time. This paper is the first in a three-part series that unifies LQG and TBM by endowing each object with the other's structure and showing that the resulting laws agree. The present work uses a form of the quantum Loewner evolution (QLE) to construct a metric on a dense subset of a $\sqrt{8/3}$-LQG sphere and to establish certain facts about the law of this metric, which are in agreement with similar facts known for TBM. The subsequent papers will show that this metric extends uniquely and continuously to the entire $\sqrt{8/3}$-LQG surface and that the resulting measure-endowed metric space is TBM., Comment: 73 pages, 14 figures

We show that the unit area Liouville quantum gravity sphere can be constructed in two equivalent ways. The first, which was introduced by the authors and Duplantier, uses a Bessel excursion measure to produce a Gaussian free field variant on the cylinder. The second uses a correlated Brownian loop and a "mating of trees" to produce a Liouville quantum gravity sphere decorated by a space-filling path. In the special case that $\gamma=\sqrt{8/3}$, we present a third equivalent construction, which uses the excursion measure of a $3/2$-stable L\'evy process (with only upward jumps) to produce a pair of trees of quantum disks that can be mated to produce a sphere decorated by SLE$_6$. This construction is relevant to a program for showing that the $\gamma=\sqrt{8/3}$ Liouville quantum gravity sphere is equivalent to the Brownian map., Comment: 63 pages and 5 figures

The Brownian map is a random sphere-homeomorphic metric measure space obtained by "gluing together" the continuum trees described by the $x$ and $y$ coordinates of the Brownian snake. We present an alternative "breadth-first" construction of the Brownian map, which produces a surface from a certain decorated branching process. It is closely related to the peeling process, the hull process, and the Brownian cactus. Using these ideas, we prove that the Brownian map is the only random sphere-homeomorphic metric measure space with certain properties: namely, scale invariance and the conditional independence of the inside and outside of certain "slices" bounded by geodesics. We also formulate a characterization in terms of the so-called L\'evy net produced by a metric exploration from one measure-typical point to another. This characterization is part of a program for proving the equivalence of the Brownian map and Liouville quantum gravity with parameter $\gamma= \sqrt{8/3}$., Comment: 134 pages and 22 figures

We study Conformal Loop Ensemble (CLE$_{\kappa}$) in doubly connected domains: annuli, the punctured disc, and the punctured plane. We restrict attention to CLE$_{\kappa}$ for which the loops are simple, i.e. $\kappa\in (8/3,4]$. In the paper "Conformal Loop Ensemble" by Sheffield and Werner, simple CLE in the unit disc is introduced and constructed as the collection of outer boundaries of outermost clusters of the Brownian loop soup. For simple CLE in the unit disc, any fixed interior point is almost surely surrounded by some loop of CLE. The gasket of the collection of loops in CLE, i.e. the set of points that are not surrounded by any loop, almost surely has Lebesgue measure zero. In the current paper, simple CLE in an annulus is constructed similarly: it is the collection of outer boundaries of outermost clusters of the Brownian loop soup conditioned on the event that there is no cluster disconnecting the two components of the boundary of the annulus. Simple CLE in the punctured disc can be viewed as simple CLE in the unit disc conditioned on the event that the origin is in the gasket. Simple CLE in the punctured plane can be viewed as simple CLE in the whole plane conditioned on the event that both the origin and infinity are in the gasket. We construct and study these three kinds of CLEs, along with the corresponding exploration processes., Comment: accepted for publication by AIHP

Given an instance $h$ of the Gaussian free field on a planar domain $D$ and a constant $\gamma \in (0,2)$, one can use various regularization procedures to make sense of the Liouville quantum gravity area measure $\mu := e^{\gamma h(z)} dz.$ It is known that the field $h$ a.s. determines the measure $\mu_h$. We show that the converse is true: namely, $h$ is measurably determined by $\mu_h$. More generally, given a random closed fractal subset $\mathcal A$ endowed with a Frostman measure $\sigma$ whose support is $\mathcal A$ (independent of $h$), a Gaussian multiplicative chaos measure $\mu_{\sigma,h}$ can be constructed. We give a mild condition on $(\mathcal A,\sigma)$ under which $\mu_{\sigma,h}$ determines $h$ restricted to $\mathcal A$, in the sense that it determines its harmonic extension off $\mathcal A$. Our condition is satisfied by the occupation measures of planar Brownian motion and SLE curves under natural parametrizations. Along the way we obtain general positive moment bounds for Gaussian multiplicative chaos. Contrary to previous results, this does not require any assumption on the underlying measure $\sigma$ such as scale invariance, and hence may be of independent interest., Comment: 26 pages. v4: substantially revised; Conjecture 1.5 of the previous version is proved under mild assumptions

There is a simple way to "glue together" a coupled pair of continuum random trees (CRTs) to produce a topological sphere. The sphere comes equipped with a measure and a space-filling curve (which describes the "interface" between the trees). We present an explicit and canonical way to embed the sphere in ${\mathbf C} \cup \{ \infty \}$. In this embedding, the measure is Liouville quantum gravity (LQG) with parameter $\gamma \in (0,2)$, and the curve is space-filling SLE$_{\kappa'}$ with $\kappa' = 16/\gamma^2$. Achieving this requires us to develop an extensive suite of tools for working with LQG surfaces. We explain how to conformally weld so-called "quantum wedges" to obtain new quantum wedges of different weights. We construct finite-volume quantum disks and spheres of various types, and give a Poissonian description of the set of quantum disks cut off by a boundary-intersecting SLE$_{\kappa}(\rho)$ process with $\kappa \in (0,4)$. We also establish a L\'evy tree description of the set of quantum disks to the left (or right) of an SLE$_{\kappa'}$ with $\kappa' \in (4,8)$. We show that given two such trees, sampled independently, there is a.s. a canonical way to "zip them together" and recover the SLE$_{\kappa'}$. The law of the CRT pair we study was shown in an earlier paper to be the scaling limit of the discrete tree/dual-tree pair associated to an FK-decorated random planar map (RPM). Together, these results imply that FK-decorated RPM scales to CLE-decorated LQG in a certain "tree structure" topology., Comment: 209 pages, approximately 60 figures; revised

We survey the properties of the log-correlated Gaussian field (LGF), which is a centered Gaussian random distribution (generalized function) $h$ on $\mathbb R^d$, defined up to a global additive constant. Its law is determined by the covariance formula $$\mathrm{Cov}\bigl[ (h, \phi_1), (h, \phi_2) \bigr] = \int_{\mathbb R^d \times \mathbb R^d} -\log|y-z| \phi_1(y) \phi_2(z)dydz$$ which holds for mean-zero test functions $\phi_1, \phi_2$. The LGF belongs to the larger family of fractional Gaussian fields obtained by applying fractional powers of the Laplacian to a white noise $W$ on $\mathbb R^d$. It takes the form $h = (-\Delta)^{-d/4} W$. By comparison, the Gaussian free field (GFF) takes the form $(-\Delta)^{-1/2} W$ in any dimension. The LGFs with $d \in \{2,1\}$ coincide with the 2D GFF and its restriction to a line. These objects arise in the study of conformal field theory and SLE, random surfaces, random matrices, Liouville quantum gravity, and (when $d=1$) finance. Higher dimensional LGFs appear in models of turbulence and early-universe cosmology. LGFs are closely related to cascade models and Gaussian branching random walks. We review LGF approximation schemes, restriction properties, Markov properties, conformal symmetries, and multiplicative chaos applications., Comment: 24 pages, 2 figures

We discuss a family of random fields indexed by a parameter $s\in \mathbb{R}$ which we call the fractional Gaussian fields, given by \[ \mathrm{FGF}_s(\mathbb{R}^d)=(-\Delta)^{-s/2} W, \] where $W$ is a white noise on $\mathbb{R}^d$ and $(-\Delta)^{-s/2}$ is the fractional Laplacian. These fields can also be parameterized by their Hurst parameter $H = s-d/2$. In one dimension, examples of $\mathrm{FGF}_s$ processes include Brownian motion ($s = 1$) and fractional Brownian motion ($1/2 < s < 3/2$). Examples in arbitrary dimension include white noise ($s = 0$), the Gaussian free field ($s = 1$), the bi-Laplacian Gaussian field ($s = 2$), the log-correlated Gaussian field ($s = d/2$), L\'evy's Brownian motion ($s = d/2 + 1/2$), and multidimensional fractional Brownian motion ($d/2 < s < d/2 + 1$). These fields have applications to statistical physics, early-universe cosmology, finance, quantum field theory, image processing, and other disciplines. We present an overview of fractional Gaussian fields including covariance formulas, Gibbs properties, spherical coordinate decompositions, restrictions to linear subspaces, local set theorems, and other basic results. We also define a discrete fractional Gaussian field and explain how the $\mathrm{FGF}_s$ with $s \in (0,1)$ can be understood as a long range Gaussian free field in which the potential theory of Brownian motion is replaced by that of an isotropic $2s$-stable L\'evy process., Comment: 73 pages

What is the scaling limit of diffusion limited aggregation (DLA) in the plane? This is an old and famously difficult question. One can generalize the question in two ways: first, one may consider the {\em dielectric breakdown model} $\eta$-DBM, a generalization of DLA in which particle locations are sampled from the $\eta$-th power of harmonic measure, instead of harmonic measure itself. Second, instead of restricting attention to deterministic lattices, one may consider $\eta$-DBM on random graphs known or believed to converge in law to a Liouville quantum gravity (LQG) surface with parameter $\gamma \in [0,2]$. In this generality, we propose a scaling limit candidate called quantum Loewner evolution, QLE$(\gamma^2, \eta)$. QLE is defined in terms of the radial Loewner equation like radial SLE, except that it is driven by a measure valued diffusion $\nu_t$ derived from LQG rather than a multiple of a standard Brownian motion. We formalize the dynamics of $\nu_t$ using an SPDE. For each $\gamma \in (0,2]$, there are two or three special values of $\eta$ for which we establish the existence of a solution to these dynamics and explicitly describe the stationary law of $\nu_t$. We also explain discrete versions of our construction that relate DLA to loop-erased random walk and the Eden model to percolation. A certain "reshuffling" trick (in which concentric annular regions are rotated randomly, like slot machine reels) facilitates explicit calculation. We propose QLE$(2,1)$ as a scaling limit for DLA on a random spanning-tree-decorated planar map, and QLE$(8/3,0)$ as a scaling limit for the Eden model on a random triangulation. We propose using QLE$(8/3,0)$ to endow pure LQG with a distance function, by interpreting the region explored by a branching variant of QLE$(8/3,0)$, up to a fixed time, as a metric ball in a random metric space., Comment: 132 pages, approximately 100 figures and computer simulations