arXiv subjects

Scott Sheffield

Publications and source records attributed to Scott Sheffield.

At least 19 recordsLinked to original sources

arXiv2025

Dynamical approach to area law for lattice Yang-Mills

In this note, we observe that the dynamical approach to lattice Yang-Mills set forth in [SZZ23] may also be applied to prove Wilson's area law in the 't Hooft regime of parameters. The main point is to verify the mass gap condition from [DF80], from which area law directly follows. Our results apply for gauge groups $G \in \{\mathrm{U}(N), \mathrm{SU}(N), \mathrm{SO}(2N)\}$, which all have nontrivial center (which is one of the key assumptions in [DF80]).

math.PR↗

arXiv2025

Expanded regimes of area law for lattice Yang-Mills theories

We extend the parameter regimes for which area law is proven for pure $\mathrm{U}(N)$ lattice Yang-Mills theories, in particular when $N$ is large. This improves on a classical result of Osterwalder-Seiler from 1978. To do so, we view the master loop equation as a linear inhomogeneous equation for Wilson string expectations, and then prove an a priori bound for solutions to the equation. The main novelty is in how we deal with the merger term in the master loop equation. This is done by introducing a truncated model for which the merger term is unproblematic, and then showing that the truncated model well approximates the original model.

math.PR↗

arXiv2024

Fractional Gaussian forms and gauge theory: an overview

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.

math.PR↗

arXiv2023

Random surfaces and lattice Yang-Mills

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.

math.PR↗

arXiv2023

Sums of GUE matrices and concentration of hives from correlation decay of eigengaps

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.

math.PR↗

arXiv2023

Scaling limits of planar maps under the Smith embedding

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).

math.PR↗

arXiv2023

Wilson loop expectations as sums over surfaces on the plane

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.

math.PR↗

arXiv2023

Large deviations for the 3D dimer model

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.

math.PR↗

arXiv2023

Brownian loops on non-smooth surfaces and the Polyakov-Alvarez formula

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.

math.PR↗

arXiv2022

What is a random 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?

math.PR↗

arXiv2021

Large deviations for random hives and the spectrum of the sum of two random matrices

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.

math.PR↗

arXiv2020

Geodesics and metric ball boundaries in Liouville quantum gravity

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.

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

Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

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.

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

Simple Conformal Loop Ensembles on Liouville Quantum Gravity

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.

math.PR↗

arXiv2018

Delocalization of uniform graph homomorphisms from $\mathbb{Z}^2$ to $\mathbb{Z}$

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.

math.PR↗

arXiv2018

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 $\epsilon \to 0$ 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.

math.PR↗