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Article
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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Article
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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Article
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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Article
Delocalization of Uniform Graph Homomorphisms from \({\mathbb {Z}}^2\) to \({\mathbb {Z}}\)
Graph homomorphisms from the \({\mathbb {Z}}^d\) Z ...
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Article
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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Article
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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Article
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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Article
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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Article
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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Chapter
Log-correlated Gaussian Fields: An Overview
We survey the properties of the log-correlated Gaussian field (LGF), which is a centered Gaussian random distribution (generalized function) h on ℝ d , defined up to a global additiv...
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Article
Open AccessImaginary geometry I: interacting SLEs
Fix constants \(\chi >0\) χ > ...
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Article
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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Article
Power law Pólya’s urn and fractional Brownian motion
We introduce a natural family of random walks \(S_n\) on
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Article
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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Article
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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Article
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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Article
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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Chapter
Tug-of-War and the Infinity Laplacian
We consider a class of zero-sum two-player stochastic games called tug-of-war and use them to prove that every bounded real-valued Lipschitz function F on a subset Y of a length space X admits a unique absolutely...
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Chapter
Contour lines of the two-dimensional discrete Gaussian free field
The 2-dimensional massless Gaussian free field (GFF) is a 2-dimensional-time analog of Brownian motion. Just as Brownian motion is a scaling limit of simple random walks and various other 1-dimensional systems...
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Article
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 ...