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Showing 1 to 17 of 17 for “"Gaussian free field"”.
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Exploring Random Geometry with the Gaussian Free Field
… SLEκ is a curve. In Chapter 2 we introduce the Gaussian free field (GFF), a conformally invariant random surface with a domain-Markov property. We explain how to couple the GFF and an SLEκ process, in particular how a GFF can be unzipped along a reverse SLEκ to produce another GFF. We also look …
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Conformal loop ensembles and the Gaussian free field
… invariant mathematical objects, such as the Gaussian free field, the Schramm-Loewner evolution, and the conformal loop ensemble. Just as Brownian motion is a scaling limit of discrete random walks, these objects serve as universal scaling limits of functions or paths associated with the …
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Gaussian free field, Schramm-Loewner evolution and Liouville quantum gravity
Consider an instance h of the Gaussian free field on a simply connected domain ... We study several properties of the level lines: continuity, monotonicity, reversibility and target-independence ... In the second part, we study Liouville quantum gravity(LQG).
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Extreme values of non-Gaussian fields
… the extremal behaviour of log-correlated spatial Gaussian processes has drawn a lot of attention. Among many other results, it is known for the lattice discrete Gaussian free field (DGFF) in $d=2$ as well as for general log-correlated Gaussian fields, that the limiting law of the centred maximum …
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Scaling limit of critical systems in random geometry
… we consider Schramm--Loewner evolutions, the Gaussian free field, Liouville quantum gravity and the Brownian continuum random tree. We begin by considering branching diffusions in a bounded domain $D\subset$ $R^{d}$, in which particles are killed upon hitting the boundary $\partial D$. It is …
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Superprobability on Graphs
… a continuous-time random walk to the square of a Gaussian free field. The Gaussian free field is a spin system (or sigma model) that takes values in Euclidean space; in this work, we generalise the classical isomorphism theorems to spin systems taking values in hyperbolic and spherical geometries. …
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Two-Dimensional Discrete Gaussian Model at High Temperature
The Discrete Gaussian model is a Gaussian free field on lattice restricted to take integer values. In dimension two, it was proved by the seminal work of Fröhlich-Spencer that the Discrete Gaussian model exhibits localisation-delocalisation phase transition. The phase transition is ubiquitous in …
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Random conformally covariant metrics in the plane
… "$e^{\gamma h} (dx^2+dy^2)$'' where $h$ is a Gaussian free field (GFF) on a planar domain and $\gamma \in (0,2)$. Duplantier and Sheffield constructed the $\gamma$-LQG area and boundary length measures, which fall under the framework of Kahane's Gaussian multiplicative chaos. Later, a …
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Fluctuations and mixing for planar random growth
… shape are described by a random holomorphic Gaussian field $\mathcal{F}$ on {|z| > 1}, of which we provide an explicit construction. We find that the boundary values of $\mathcal{F}$ perform an Ornstein-Uhlenbeck process on an infinite-dimensional Hilbert space, which can be characterised as …
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Random tilings : gap probabilities, local and global asymptotics
… the fluctuations of the height functions to the Gaussian Free Field in appropriate coordinates. Our main tool is a recently developed moment method for discrete particle systems.
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Random partitions and the quantum Benjamin-Ono hierarchy
… along v of the restriction to the circle of a Gaussian free field on the upper half-plane whose covariance is independent of [beta]. At [beta] = 2, our result matches Breuer-Duits' central limit theorem (2013) for Borodin's biorthogonal ensembles. Our limit theorems follow from a diagrammatic …
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Thick points of random walk and multiplicative chaos
… unrelated area of Probability theory, called Gaussian multiplicative chaos. Firstly, in two dimensions, we answer a question of Dembo, Peres, Rosen and Zeitouni and compute the number of thick points of planar random walk, assuming that the increments are symmetric and have a finite moment of …
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Homogenization of Random Media: Random Walks, Diffusions and Stochastic Interface Models
… the associated Gibbs distribution scales to a Gaussian free field. In the final chapter, we study a symmetric diffusion process in divergence form in a stationary and ergodic random environment. This is a continuum analogue of the random conductance model and similar analytical techniques are …
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Continuum Methods and Tightness Results for Non-Simple SLE and CLE: Existence, Interactions, and Metric Approximations
… in which SLE curves arise as flow lines of the Gaussian Free Field (GFF). We work with whole-plane space-filling SLEκ parameterized by Lebesgue measure, and establish uniform estimates on its behavior as κ↑8. A tightness argument allows us to take a limit in law, yielding a continuous version of …
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Macroscopic behaviour of Lipschitz random surfaces
A random field is a random function φ from the square lattice ℤᵈ to some fixed standard Borel space (E, ℰ). A random surface is a random field with the extra condition that E ∈ {ℤ, ℝ} where ℰ is the standard σ-algebra. For random surfaces, one often studies the gradient ∇φ of the random function of …
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Fine regularity properties of SLE_4 and SLE_8
… study as it has deep connections to the Gaussian free field (GFF) and Liouville quantum gravity (LQG). The first part of the thesis treats the values $\kappa$ = 4 and $\kappa$ = 8. The value $\kappa$ = 4 is special because it is the critical value at or below which SLE$_{\kappa}$ curves …
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On Gaussian multiplicative chaos and conformal field theory
… dimensions and their applications to conformal field theory (CFT). The main probabilistic objects are the Gaussian free field (GFF) and the random geometries associated to it. Especially, we are interested in Gaussian multiplicative chaos (GMC), Schramm-Loewner evolution (SLE) and Liouville CFT, …