Scott Sheffield⋆
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View article: Random surfaces and lattice Yang-Mills
Random surfaces and lattice Yang-Mills Open
We study Wilson loop expectations in lattice Yang-Mills models with a compact Lie group . Using tools recently introduced in a companion paper (see M. Park, J. Pfeffer, S. Sheffield, and P. Yu [ Wilson loop expectations as sums over surfa…
View article: Wilson loop expectations as sums over surfaces on the plane
Wilson loop expectations as sums over surfaces on the plane Open
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 som…
View article: Liouville Quantum Gravity as a Mating of Trees
Liouville Quantum Gravity as a Mating of Trees Open
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…
View article: Dynamical approach to area law for lattice Yang-Mills
Dynamical approach to area law for lattice Yang-Mills Open
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 [D…
View article: Expanded regimes of area law for lattice Yang-Mills theories
Expanded regimes of area law for lattice Yang-Mills theories Open
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 …
View article: Statistical Physics and Random Surfaces
Statistical Physics and Random Surfaces Open
This conference featured a diverse group of participants, from various career stages, to study problems in several hot topics that have grown increasingly prominent and interrelated in recent years. These included the following: (1) random…
View article: Fractional Gaussian forms and gauge theory: an overview
Fractional Gaussian forms and gauge theory: an overview Open
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 f…
View article: What is a random surface?
What is a random surface? Open
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 in…
View article: Random surfaces and lattice Yang-Mills
Random surfaces and lattice Yang-Mills Open
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 the…
View article: Sums of GUE matrices and concentration of hives from correlation decay of eigengaps
Sums of GUE matrices and concentration of hives from correlation decay of eigengaps Open
Associated to two given sequences of eigenvalues $λ_1 \geq \dots \geq λ_n$ and $μ_1 \geq \dots \geq μ_n$ is a natural polytope, the polytope of augmented hives with the specified boundary data, which is associated to sums of random Hermiti…
View article: Scaling limits of planar maps under the Smith embedding
Scaling limits of planar maps under the Smith embedding Open
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 c…
View article: Large deviations for the 3D dimer model
Large deviations for the 3D dimer model Open
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 functio…
View article: Brownian loops on non-smooth surfaces and the Polyakov-Alvarez formula
Brownian loops on non-smooth surfaces and the Polyakov-Alvarez formula Open
Let $ρ$ be compactly supported on $D \subset \mathbb R^2$. Endow $\mathbb R^2$ with the metric $e^ρ(dx_1^2 + dx_2^2)$. As $δ\to 0$ the set of Brownian loops centered in $D$ with length at least $δ$ has measure $$\frac{\text{area}(D)}{2πδ} …
View article: An invariance principle for ergodic scale-free random environments
An invariance principle for ergodic scale-free random environments Open
does the same with M and M ′ reversed.Note that the integrand is equal to e -r for each r>0 such an invariance principle for ergodic scale-free random environments 305 that no such homeomorphism exists.When we speak of random embedded latt…
View article: Liouville quantum gravity and the Brownian map II: Geodesics and continuity of the embedding
Liouville quantum gravity and the Brownian map II: Geodesics and continuity of the embedding Open
We endow the $\sqrt{8/3}$-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 natur…
View article: Large deviations for random hives and the spectrum of the sum of two random matrices
Large deviations for random hives and the spectrum of the sum of two random matrices Open
Suppose $α, β$ are Lipschitz strongly concave functions from $[0, 1]$ to $\mathbb{R}$ and $γ$ is a concave function from $[0, 1]$ to $\mathbb{R}$, such that $α(0) = γ(0) = 0$, and $α(1) = β(0) = 0$ and $β(1) = γ(1) = 0.$ For an $n \times n…
View article: The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity
The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity Open
We prove that the Tutte embeddings (a.k.a. harmonic/barycentric embeddings) of certain random planar maps converge to $γ$-Liouville quantum gravity ($γ$-LQG). Specifically, we treat mated-CRT maps, which are discretized matings of correlat…
View article: An axiomatic characterization of the Brownian map
An axiomatic characterization of the Brownian map Open
The Brownian map is a random sphere-homeomorphic metric measure space obtained by “gluing together” the continuum trees described by the and coordinates of the Brownian snake. We present an alternative “breadth-first” construction of the…
View article: Geodesics and metric ball boundaries in Liouville quantum gravity
Geodesics and metric ball boundaries in Liouville quantum gravity Open
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 $γ\in (0,2)$. We establish a strong confluence property for LQG geodesics, which…
View article: Best and worst policy control in low-prevalence SEIR
Best and worst policy control in low-prevalence SEIR Open
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…
View article: Strict Physical Distancing May Be More Efficient: A Mathematical Argument for Making Lockdowns Count
Strict Physical Distancing May Be More Efficient: A Mathematical Argument for Making Lockdowns Count Open
COVID-19 created a global public health and economic emergency. Policymakers acted quickly and decisively to contain the spread of disease through physical distancing measures. However, these measures also impact physical, mental and econo…
View article: Brownian loops and the central charge of a Liouville random surface
Brownian loops and the central charge of a Liouville random surface Open
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 appr…
View article: Simple Conformal Loop Ensembles on Liouville Quantum Gravity
Simple Conformal Loop Ensembles on Liouville Quantum Gravity Open
We show that when one draws a simple conformal loop ensemble (CLE$_κ$ for $κ\in (8/3,4)$) on an independent $\sqrtκ$-Liouville quantum gravity (LQG) surface and explores the CLE in a natural Markovian way, the quantum surfaces (e.g., corre…
View article: Laplacian determinants and random surfaces
Laplacian determinants and random surfaces Open
I will discuss how dimer models and other statistical physics models are related to Laplacian determinants, both on the discrete level and on the continuum level. In particular, I will recall the geometric meaning of the so-called zeta-reg…
View article: Non-simple SLE curves are not determined by their range
Non-simple SLE curves are not determined by their range Open
We show that when observing the range of a chordal SLE _\kappa curve for \kappa \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)…