arXiv subjects

Scott Sheffield

Publications and source records attributed to Scott Sheffield.

At least 37 records · Page 2Linked to original sources

arXiv2020

Liouville quantum gravity and the Brownian map III: the conformal structure is determined

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 Möbius 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.

math.PR↗

arXiv2020

Best and worst policy control in low-prevalence SEIR

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.

math.OC↗

arXiv2020

Equivalence of Liouville measure and Gaussian free field

Given an instance $h$ of the Gaussian free field on a planar domain $D$ and a constant $γ\in (0,2)$, one can use various regularization procedures to make sense of the Liouville quantum gravity area measure $μ:= e^{γh(z)} dz.$ It is known that the field $h$ a.s. determines the measure $μ_h$. We show that the converse is true: namely, $h$ is measurably determined by $μ_h$. More generally, given a random closed fractal subset $\mathcal A$ endowed with a Frostman measure $σ$ whose support is $\mathcal A$ (independent of $h$), a Gaussian multiplicative chaos measure $μ_{σ,h}$ can be constructed. We give a mild condition on $(\mathcal A,σ)$ under which $μ_{σ,h}$ 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 $σ$ such as scale invariance, and hence may be of independent interest.

math.PR↗

arXiv2020

Liouville quantum gravity as a mating of trees

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 $γ\in (0,2)$, 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 $κ\in (0,4)$. We also establish a Lévy tree description of the set of quantum disks to the left (or right) of an SLE$_{κ'}$ with $κ' \in (4,8)$. We show that given two such trees, sampled independently, there is a.s. a canonical way to "zip them together" and recover the SLE$_{κ'}$. 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.

math.PR↗

arXiv2020

Brownian loops and the central charge of a Liouville random surface

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 $\mathbf{c}>1$.

math.PR↗

arXiv2020

An axiomatic characterization of the Brownian map

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 Liouville quantum gravity with parameter $γ= \sqrt{8/3}$.

math.PR↗

arXiv2019

The Tutte embedding of the Poisson-Voronoi tessellation of the Brownian disk converges to $\sqrt{8/3}$-Liouville quantum gravity

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 $ε\to 0$ limit of simple random walk on $ε\mathbb Z^2$? We answer this question affirmatively by showing that Brownian motion on a Brownian surface is (up to time change) the $λ\to \infty$ limit of simple random walk on the Voronoi tessellation induced by a Poisson point process whose intensity is $λ$ times the associated area measure. Among other things, this implies that as $λ\to \infty$ 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.

math.PR↗

arXiv2019

Liouville quantum gravity and the Brownian map I: The QLE(8/3,0) metric

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 $γ= \sqrt{8/3}$, 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.

math.PR↗

arXiv2019

Harmonic functions on mated-CRT maps

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 $γ$-Liouville quantum gravity (LQG) for $γ\in (0,2)$ if we take the correlation to be $-\cos(πγ^2/4)$. 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.

math.PR↗

arXiv2018

Non-simple SLE curves are not determined by their range

We show that when observing the range of a chordal SLE$_κ$ curve for $κ\in (4,8)$, 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$_κ$ for $κ\in (4,8)$ are not determined by the CLE$_κ$ gasket. (ii) The continuum percolation interfaces defined in the fractal carpets of conformal loop ensembles CLE$_κ$ for $κ\in (8/3, 4)$ (we defined these percolation interfaces in previous work, and showed there that they are SLE$_{16/κ}$ curves) are not determined by the CLE$_κ$ carpet that they are defined in.

math.PR↗

arXiv2018

Liouville quantum gravity spheres as matings of finite-diameter trees

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 $γ=\sqrt{8/3}$, we present a third equivalent construction, which uses the excursion measure of a $3/2$-stable Lévy 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 $γ=\sqrt{8/3}$ Liouville quantum gravity sphere is equivalent to the Brownian map.

math.PR↗

arXiv2018

Scaling limits of the Schelling model

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 $h-ϕ$ at zero (taken over all $1$-Lipschitz functions $ϕ$) 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.

math.PR↗

arXiv2017

CLE percolations

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($κ$) with $κ$ between 8/3 and 4) and the corresponding CLE($κ'$) where $κ':=16/κ$, 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($κ$) as follows: sample a CLE($κ'$), 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($κ'$) loops as interfaces of a continuum analog of critical percolation within a CLE($κ)$ 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$(κ;ρ)$ 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 $κ$ between 2 and 4.

math.PR↗

arXiv2017

Fixation for Distributed Clustering Processes

We study a discrete-time resource flow in $Z^d$, where wealthier vertices attract the resources of their less rich neighbors. For any translation-invariant probability distribution of initial resource quantities, we prove that the flow at each vertex terminates after finitely many steps. This answers (a generalized version of) a question posed by van den Berg and Meester in 1991. The proof uses the mass-transport principle and extends to other graphs.

math.PR↗

arXiv2017

The six-vertex model and Schramm-Loewner evolution

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$_κ$, Schramm--Loewner evolution with parameter $κ$, where $4<κ\leq 12+8\sqrt{2}$. For square ice, $κ=12$. At the "free-fermion point" of the 6-vertex model, $κ=8+4\sqrt{3}$. These unusual values lie outside the classical interval $2\le κ\le 8$.

cond-mat.stat-mech↗

arXiv2017

Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees

We establish existence and uniqueness for Gaussian free field flow lines started at {\em 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 {\em 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\in [0,4]) to all κ\in [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_κfor some κ\in (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\in (4, \infty). 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' \geq 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' \in (4,8), and describe the laws of the boundaries of \SLE_{κ'} processes stopped at stopping times.

math.PR↗

arXiv2017

Field-measure correspondence in Liouville quantum gravity almost surely commutes with all conformal maps simultaneously

In Liouville quantum gravity (or $2d$-Gaussian multiplicative chaos) one seeks to define a measure $μ^h = e^{γh(z)} dz$ 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 $h_ε(z)$ be the average value of $h$ on the circle of radius $ε$ centered at $z$ (or an analogous average defined using a bump function supported inside that circle) and then write $μ^h = \lim_{ε\to 0} ε^{\frac{γ^2}{2}} e^{γh_ε(z)} dz$. If $ϕ: \tilde D \to D$ is a conformal map, one can write $\tilde h = h \circ ϕ+ Q \log |ϕ'|$, where $Q = 2/γ+ γ/2$. The measure $μ^{\tilde h}$ on $\tilde D$ is then a.s.\ equivalent to the pullback via $ϕ^{-1}$ of the measure $μ^h$ on $D$. Interestingly, although this a.s.\ holds for each \textit{given} $ϕ$, nobody has ever proved that it a.s.\ holds \textit {simultaneously} for all possible $ϕ$. 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 $(D,μ^{h})$ 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.

math.PR↗

arXiv2016

Bipolar orientations on planar maps and SLE$_{12}$

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 $κ=12$ (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.

math.PR↗