Scott Sheffield

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.

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.

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.

We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble $$\hbox {CLE}_{\kappa '}$$CLE κ ′ for $$\kappa '$$κ ′ in (4, 8) that is drawn on an independent $$\gamma $$γ -LQG surface for $$\gamma ^2=16/\kappa '$$γ 2 = 16 / κ ′ . The results are similar in flavor to the ones from our companion paper dealing with $$\hbox {CLE}_{\kappa }$$CLE κ for $$\kappa $$κ 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 $$\hbox {CLE}_{\kappa '}$$CLE κ ′ 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: In our paper entitled “CLE Percolations” described the law of interfaces obtained when coloring the loops of a $$\hbox {CLE}_{\kappa '}$$CLE κ ′ independently into two colors with respective probabilities p and $$1-p$$1 - p . This description was complete up to one missing parameter $$\rho $$ρ . The results of the present paper about CLE on LQG allow us to determine its value in terms of p and $$\kappa '$$κ ′ . It shows in particular that $$\hbox {CLE}_{\kappa '}$$CLE κ ′ and $$\hbox {CLE}_{16/\kappa '}$$CLE 16 / κ ′ are related via a continuum analog of the Edwards-Sokal coupling between $$\hbox {FK}_q$$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 $$q=4\cos ^2(4\pi / \kappa ')$$q = 4 cos 2 ( 4 π / κ ′ ) . This provides further evidence for the long-standing belief that $$\hbox {CLE}_{\kappa '}$$CLE κ ′ and $$\hbox {CLE}_{16/\kappa '}$$CLE 16 / κ ′ represent the scaling limits of $$\hbox {FK}_q$$FK q percolation and the q-Potts model when q and $$\kappa '$$κ ′ are related in this way. Another consequence of the formula for $$\rho (p,\kappa ')$$ρ ( p , κ ′ ) 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.

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.

Previous works in this series have shown that an instance of a $$\sqrt{8/3}$$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 Möbius transformation). In other words, an instance of TBM has a canonical conformal structure. The conclusion is that TBM and the $$\sqrt{8/3}$$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 $$\sqrt{8/3}$$8 / 3 -LQG surfaces with other topologies.