112 research outputs found
Scale-invariant random geometry from mating of trees: a numerical study
The search for scale-invariant random geometries is central to the Asymptotic
Safety hypothesis for the Euclidean path integral in quantum gravity. In an
attempt to uncover new universality classes of scale-invariant random
geometries that go beyond surface topology, we explore a generalization of the
mating of trees approach introduced by Duplantier, Miller and Sheffield. The
latter provides an encoding of Liouville Quantum Gravity on the 2-sphere
decorated by a certain random space-filling curve in terms of a two-dimensional
correlated Brownian motion, that can be viewed as describing a pair of random
trees. The random geometry of Liouville Quantum Gravity can be conveniently
studied and simulated numerically by discretizing the mating of trees using the
Mated-CRT maps of Gwynne, Miller and Sheffield. Considering higher-dimensional
correlated Brownian motions, one is naturally led to a sequence of non-planar
random graphs generalizing the Mated-CRT maps that may belong to new
universality classes of scale-invariant random geometries. We develop a
numerical method to efficiently simulate these random graphs and explore their
possible scaling limits through distance measurements, allowing us in
particular to estimate Hausdorff dimensions in the two- and three-dimensional
setting. In the two-dimensional case these estimates accurately reproduce
previous known analytic and numerical results, while in the three-dimensional
case they provide a first window on a potential three-parameter family of new
scale-invariant random geometries.Comment: 28 pages, 16 figure
Models and Methods for Random Fields in Spatial Statistics with Computational Efficiency from Markov Properties
The focus of this work is on the development of new random field models and methods suitable for the analysis of large environmental data sets. A large part is devoted to a number of extensions to the newly proposed Stochastic Partial Differential Equation (SPDE) approach for representing Gaussian fields using Gaussian Markov Random Fields (GMRFs). The method is based on that Gaussian Matérn field can be viewed as solutions to a certain SPDE, and is useful for large spatial problems where traditional methods are too computationally intensive to use. A variation of the method using wavelet basis functions is proposed and using a simulation-based study, the wavelet approximations are compared with two of the most popular methods for efficient approximations of Gaussian fields. A new class of spatial models, including the Gaussian Matérn fields and a wide family of fields with oscillating covariance functions, is also constructed using nested SPDEs. The SPDE method is extended to this model class and it is shown that all desirable properties are preserved, such as computational efficiency, applicability to data on general smooth manifolds, and simple non-stationary extensions. Finally, the SPDE method is extended to a larger class of non-Gaussian random fields with Matérn covariance functions, including certain Laplace Moving Average (LMA) models. In particular it is shown how the SPDE formulation can be used to obtain an efficient simulation method and an accurate parameter estimation technique for a LMA model. A method for estimating spatially dependent temporal trends is also developed. The method is based on using a space-varying regression model, accounting for spatial dependency in the data, and it is used to analyze temporal trends in vegetation data from the African Sahel in order to find regions that have experienced significant changes in the vegetation cover over the studied time period. The problem of estimating such regions is investigated further in the final part of the thesis where a method for estimating excursion sets, and the related problem of finding uncertainty regions for contour curves, for latent Gaussian fields is proposed. The method is based on using a parametric family for the excursion sets in combination with Integrated Nested Laplace Approximations (INLA) and an importance sampling-based algorithm for estimating joint probabilities
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Scaling limit of critical systems in random geometry
This thesis focusses on the properties of, and relationships between, several fundamental objects arising from critical physical models. In particular, 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 , in which particles are killed upon hitting the boundary . It is known that such a system displays a phase transition in the branching rate: if it exceeds a critical value, the population will no longer become extinct almost surely. We prove that at criticality, under mild assumptions on the branching mechanism and diffusion, the genealogical tree associated with the process will converge to the Brownian CRT.
Next, we move on to study Gaussian multiplicative chaos. This is the rigorous framework that allows one to make sense of random measures built from rough Gaussian fields, and again there is a parameter associated with the model in which a phase transition occurs. We prove a uniqueness and convergence result for approximations to these measures at criticality.
From this point onwards we restrict our attention to two-dimensional models. First,
we give an alternative, ``non-Gaussian" construction of Liouville quantum gravity (a special case of Gaussian multiplicative chaos associated with the 2-dimensional Gaussian free field), that is motivated by the theory of multiplicative cascades. We prove that the Liouville (GMC) measures associated with the Gaussian free field can be approximated using certain sequences of ``local sets" of the field. This is a particularly natural construction as it is both local and conformally invariant. It includes the case of nested CLE, when it is coupled with the GFF as its set of ``level lines".
Finally, we consider this level line coupling more closely, now when it is between SLE and the GFF. We prove that level lines can be defined for the GFF with a wide range of boundary conditions, and are given by SLE-type curves. As a consequence, we extend the definition of SLE to the case of a continuum of force points
Mating of trees for random planar maps and Liouville quantum gravity: a survey
We survey the theory and applications of mating-of-trees bijections for
random planar maps and their continuum analog: the mating-of-trees theorem of
Duplantier, Miller, and Sheffield (2014). The latter theorem gives an encoding
of a Liouville quantum gravity (LQG) surface decorated by a Schramm-Loewner
evolution (SLE) curve in terms of a pair of correlated linear Brownian motions.
We assume minimal familiarity with the theory of SLE and LQG.
Mating-of-trees theory enables one to reduce problems about SLE and LQG to
problems about Brownian motion and leads to deep rigorous connections between
random planar maps and LQG. Applications discussed in this article include
scaling limit results for various functionals of decorated random planar maps,
estimates for graph distances and random walk on (not necessarily uniform)
random planar maps, computations of the Hausdorff dimensions of sets associated
with SLE, scaling limit results for random planar maps conformally embedded in
the plane, and special symmetries for -LQG which allow one to prove
its equivalence with the Brownian map.Comment: 68 pages, 12 figure
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