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 C ∪ {∞}. In this embedding, the measure is Liouville quantum gravity (LQG) with parameter γ ∈ (0, 2), and the curve is space-filling SLE κ with κ = 16/γ 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 κ (ρ) process with κ ∈ (0, 4). We also establish a Lévy tree description of the set of quantum disks to the left (or right) of an SLE κ with κ ∈ (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 κ .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.

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 Lévy 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 the Liouville quantum gravity sphere with parameter γ = 8/3.

We show that when one draws a simple conformal loop ensemble (CLEκ for κ ∈ (8/3, 4)) on an independent √ κ-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/κ)-stable processes, and labeled branching trees. The ratio between positive and negative jump intensities of these processes turns out to be − cos(4π/κ), 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 of this result 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 SLEκ(κ − 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 Lévy-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 will be the topic of another paper. Contents 1. Introduction 1 2. Background 11 3. Exploring CLEs on quantum half-planes 17 4. CLE explorations of quantum disks 23 5. The natural quantum measure in the CLE carpet 27 Appendix A. Proof of Lemma 3.3 34 References 36

We endow the 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 S 2 . Previous work in this series used quantum Loewner evolution (QLE) to construct a metric d Q on a countable dense subset of S 2 . Here we show that d Q a.s. extends uniquely and continuously to a metric d Q on all of S 2 . Letting d denote the Euclidean metric on S 2 , we show that the identity map between (S 2 , d) and (S 2 , d Q ) is a.s. Hölder continuous in both directions. We establish several other properties of (S 2 , d 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 (S 2 , d Q ) geodesics.

We show that the measure on markings of Z d n , d ≥ 3, with elements of {0, 1} given by i.i.d. fair coin flips on the range R of a random walk X run until time T and 0 otherwise becomes indistinguishable from the uniform measure on such markings at the threshold T = 1 2 Tcov(Z d n ). As a consequence of our methods, we show that the total variation mixing time of the random walk on the lamplighter graph Z2 ≀ Z d n , d ≥ 3, has a cutoff with threshold 1 2 Tcov(Z d n ). We give a general criterion under which both of these results hold; other examples for which this applies include bounded degree expander families, the intersection of an infinite supercritical percolation cluster with an increasing family of balls, the hypercube and the Caley graph of the symmetric group generated by transpositions. The proof also yields precise asymptotics for the decay of correlation in the uncovered set.

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 whole-plane 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.

Two distinct patterns of sweet taste liking have been described: one showing a peak liking response in the mid-range of sucrose concentrations and the other showing a monotonic liking response at progressively higher sucrose concentrations. Classification of these patterns has been somewhat arbitrary. In this report, we analyzed patterns of sweet taste liking in a pilot study with 26 adults including 14 women and 12 men, 32.6 ± 14.5 years of age with body mass index 26.4 ± 5.1 kg/m2 (mean ± SD). Sweet taste liking was measured for 10 levels of sucrose solutions (0.035 M to 1.346 M). Participants rated their liking of each solution using a visual analog scale with 0 indicating strongly disliking and 100 strongly liking. The cluster analysis demonstrated two distinct groups: 13 liked relatively low sucrose concentrations and liked high sucrose concentrations less, and 13 liked high sucrose concentrations greatly. If we use the 0.598 M sucrose solution alone and a cutoff liking score of 50, we can distinguish the two clusters with high sensitivity (100%) and specificity (100%). If validated in additional studies, this simple tool may help us to better understand eating behaviors and the impact of sweet taste liking on nutrition-related disorders.

We show that when observing the range of a chordal SLEκ curve for κ ∈ (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 CLEκ for κ ∈ (4, 8) are not determined by the CLEκ gasket. (ii) The continuum percolation interfaces defined in the fractal carpets of conformal loop ensembles CLEκ for κ ∈ (8/3, 4) (we defined these percolation interfaces in earlier work, where we also showed there that they are SLE 16/κ curves) are not determined by the CLEκ carpet that they are defined in. arXiv:1609.04799v4 [math.PR]