19 Result(s)
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Scott Sheffield
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Sums of GUE matrices and concentration of hives from correlation decay of eigengaps
Associated to two given sequences of eigenvalues \(\lambda _1 \ge \cdots \ge \lambda _n\) ...
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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
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Open AccessNon-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 $$\hbox {CLE...
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Delocalization of Uniform Graph Homomorphisms from \({\mathbb {Z}}^2\) to \({\mathbb {Z}}\)
Graph homomorphisms from the \({\mathbb {Z}}^d\) Z ...
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Open AccessLiouville quantum gravity and the Brownian map III: the conformal structure is determined
Previous works in this series have shown that an instance of a \(\sqrt{8/3}\) ...
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Open AccessThe 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 \(\sqrt{8/3}\)8/3-Liouville qu...
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Scaling limits of the Schelling model
The Schelling model of segregation, 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 o...
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Open AccessLiouville quantum gravity and the Brownian map I: the \(\mathrm{QLE}(8/3,0)\) 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 \(\gamma \)γ, and it has long been believed that ...
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Open AccessImaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees
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 wa...
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Open AccessImaginary geometry I: interacting SLEs
Fix constants \(\chi >0\) χ > ...
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Renormalization of Critical Gaussian Multiplicative Chaos and KPZ Relation
Gaussian Multiplicative Chaos is a way to produce a measure on \({\mathbb{R}^d}\) ...
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Power law Pólya’s urn and fractional Brownian motion
We introduce a natural family of random walks \(S_n\) on
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A contour line of the continuum Gaussian free field
Consider an instance \(h\) of the Gaussian free field on a simply connected planar domain
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Absolutely minimal Lipschitz extension of tree-valued map**s
We prove that every Lipschitz function from a subset of a locally compact length space to a metric tree has a unique absolutely minimal Lipschitz extension (AMLE). We relate these extensions to a stochastic ga...
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Liouville quantum gravity and KPZ
Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2π)−1∫ D ∇h(z)⋅∇h(z)dz, and a constant 0≤γ<2. The Liouville quantum grav...
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The covariant measure of SLE on the boundary
We construct a natural measure μ supported on the intersection of a chordal SLE(κ) curve γ with \({\mathbb{R}}\) , in the range...
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Conformal Radii for Conformal Loop Ensembles
The conformal loop ensembles CLE κ , defined for 8/3 ≤ κ ≤ 8, are random collections of loops in a planar domain which are conjectured scaling limits of the O(n) loop models. We ...
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Contour lines of the two-dimensional discrete Gaussian free field
We prove that the chordal contour lines of the discrete Gaussian free field converge to forms of SLE(4). Specifically, there is a constant λ > 0 such that when h is an interpolation of the discrete Gaussian free ...
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Gaussian free fields for mathematicians
The d-dimensional Gaussian free field (GFF), also called the (Euclidean bosonic) massless free field, is a d-dimensional-time analog of Brownian motion. Just as Brownian motion is the limit of the simple random w...
