Fix constants χ > 0 and θ ∈ [0, 2π), and let h be an instance of the Gaussian free field on a planar domain. We study flow lines of the vector field e i(h/χ +θ) starting at a fixed boundary point of the domain. Letting θ vary, one obtains a family of curves that look locally like SLE κ processes with κ ∈ (0, 4) (where2 ), which we interpret as the rays of a random geometry with purely imaginary curvature. We extend the fundamental existence and uniqueness results about these paths to the case that the paths intersect the boundary. We also show that flow lines of different angles cross each other at most once but (in contrast to what happens when h is smooth) may bounce off of each other after crossing. Flow lines of the same angle started at different points merge into each other upon intersecting, forming a tree structure. We construct so-called counterflow lines (SLE 16/κ ) within the same geometry using ordered "light cones" of points accessible by angle-restricted trajectories and develop a robust theory of flow and counterflow line interaction. The theory leads to new results about SLE. For example, we prove that SLE κ (ρ) processes are almost surely continuous random curves, even when they intersect the boundary, and establish Duplantier duality for general SLE 16/κ (ρ) processes. Mathematics Subject Classification 60J67B Jason Miller

We establish existence and uniqueness for Gaussian free field flow lines started at interior points of a planar domain. We interpret these as rays of a random geometry with imaginary curvature and describe the way distinct rays intersect each other and the boundary. Previous works in this series treat rays started at boundary points and use Gaussian free field machinery to determine which chordal SLE κ (ρ 1 ; ρ 2 ) processes are time-reversible when κ < 8. Here we extend these results to wholeplane SLE κ (ρ) and establish continuity and transience of these paths. In particular, we extend ordinary whole-plane SLE reversibility (established by Zhan for κ ∈ [0, 4]) to all κ ∈ [0, 8]. We also show that the rays of a given angle (with variable starting point) form a space-filling planar tree. Each branch is a form of SLE κ for some κ ∈ (0, 4), and the curve that traces the tree in the natural order (hitting x before y if the branch from x is left of the branch from y) is a space-filling form of SLE κ where κ := 16/κ ∈ (4, ∞). By varying the boundary data we obtain, for each κ > 4, a family of space-filling variants of SLE κ (ρ) whose time reversals belong to the same family. When κ ≥ 8, ordinary SLE κ belongs to this family, and our result shows that its time-reversal is SLE κ (κ /2−4; κ /2−4). As applications of this theory, we obtain the local finiteness of CLE κ , for κ ∈ (4, 8), and describe the laws of the boundaries of SLE κ processes stopped at stopping times. Mathematics Subject Classification 60J67B Jason Miller

Conformal loop ensembles (CLEs) 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 loop is a canonical random connected fractal set -a random and conformally invariant analog of the Sierpinski carpet or gasket.In the present paper, we derive a direct relationship between the CLEs with simple loops (CLE κ for κ ∈ (8/3, 4), whose loops are Schramm's SLE κ -type curves) and the corresponding CLEs with nonsimple loops (CLE κ with κ := 16/κ ∈ (4, 6), whose loops are SLE κ -type curves). This correspondence 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 noninteger q.Like its discrete analog, our continuum correspondence has two directions. First, we show that for each κ ∈ (8/3, 4), one can construct a variant of CLE κ as follows: start with an instance of CLE κ , then use a biased coin to independently color each CLE κ loop in 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 κ loops as interfaces of a continuum analog of critical Bernoulli percolation within CLE κ carpets -this is the first construction of continuum percolation on a fractal planar domain. It extends and generalizes the continuum percolation on open domains defined by SLE 6 and CLE 6 .These constructions allow us to prove several conjectures made by the second author and provide new and perhaps surprising interpretations of the relationship between CLEs and the Gaussian free field. Along the way, we obtain new results about generalized SLE κ (ρ) curves for ρ < −2, such as their decomposition into collections of SLE κ -type 'loops' hanging off of SLE κ -type 'trunks', and vice versa (exchanging κ and κ ). We also 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

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 γ, and it has long been believed that when γ = 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 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 8/3-LQG surface and that the resulting measure-endowed metric space is TBM.We have benefited from conversations about this work with many people, a partial list of whom includes

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We show that for each $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , there is a unique metric (i.e., distance function) associated with $$\gamma $$ γ -Liouville quantum gravity (LQG). More precisely, we show that for the whole-plane Gaussian free field (GFF) h, there is a unique random metric $$D_h$$ D h associated with the Riemannian metric tensor “$$e^{\gamma h} (dx^2 + dy^2)$$ e γ h ( d x 2 + d y 2 ) ” on $${\mathbb {C}}$$ C which is characterized by a certain list of axioms: it is locally determined by h and it transforms appropriately when either adding a continuous function to h or applying a conformal automorphism of $$\mathbb {C}$$ C (i.e., a complex affine transformation). Metrics associated with other variants of the GFF can be constructed using local absolute continuity. The $$\gamma $$ γ -LQG metric can be constructed explicitly as the scaling limit of Liouville first passage percolation (LFPP), the random metric obtained by exponentiating a mollified version of the GFF. Earlier work by Ding et al. (Tightness of Liouville first passage percolation for $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , 2019. arXiv:1904.08021) showed that LFPP admits non-trivial subsequential limits. This paper shows that the subsequential limit is unique and satisfies our list of axioms. In the case when $$\gamma = \sqrt{8/3}$$ γ = 8 / 3 , our metric coincides with the $$\sqrt{8/3}$$ 8 / 3 -LQG metric constructed in previous work by Miller and Sheffield, which in turn is equivalent to the Brownian map for a certain variant of the GFF. For general $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , we conjecture that our metric is the Gromov–Hausdorff limit of appropriate weighted random planar map models, equipped with their graph distance. We include a substantial list of open problems.

Let $U\subseteq\mathbf{C}$ be a bounded domain with smooth boundary and let $F$ be an instance of the continuum Gaussian free field on $U$ with respect to the Dirichlet inner product $\int_U\nabla f(x)\cdot \nabla g(x)\,dx$. The set $T(a;U)$ of $a$-thick points of $F$ consists of those $z\in U$ such that the average of $F$ on a disk of radius $r$ centered at $z$ has growth $\sqrt{a/\pi}\log\frac{1}{r}$ as $r\to 0$. We show that for each $0\leq a\leq2$ the Hausdorff dimension of $T(a;U)$ is almost surely $2-a$, that $\nu_{2-a}(T(a;U))=\infty$ when $02$. Furthermore, we prove that $T(a;U)$ is invariant under conformal transformations in an appropriate sense. The notion of a thick point is connected to the Liouville quantum gravity measure with parameter $\gamma$ given formally by $\Gamma(dz)=e^{\sqrt{2\pi}\gamma F(z)}\,dz$ considered by Duplantier and Sheffield.Comment: Published in at http://dx.doi.org/10.1214/09-AOP498 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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 dielectric breakdown model η-DBM, a generalization of DLA in which particle locations are sampled from the η-th power of harmonic measure, instead of harmonic measure itself. Second, instead of restricting attention to deterministic lattices, one may consider η-DBM on random graphs known or believed to converge in law to a Liouville quantum gravity (LQG) surface with parameter γ ∈ [0, 2].In this generality, we propose a scaling limit candidate called quantum Loewner evolution, QLE(γ 2 , η). QLE is defined in terms of the radial Loewner equation like radial SLE, except that it is driven by a measure valued diffusion ν t derived from LQG rather than a multiple of a standard Brownian motion. We formalize the dynamics of ν t using an SPDE. For each γ ∈ (0, 2], there are two or three special values of η for which we establish the existence of a solution to these dynamics and explicitly describe the stationary law of ν 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 spanningtree-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.