A level line of the Gaussian free field with measure-valued boundary conditions

In this article, we construct samples of Schramm-Loewner-evolution-like curves out of samples of the conformal loop ensemble and Poisson point processes of Brownian excursions. We show that the laws of these curves depend continuously on the intensity measure of the Brownian excursions. Using such a construction of curves, we extend the notion of level lines of the Gaussian free field to the case where the boundary condition is measure-valued.     

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 field on a Jordan domain—with boundary values −λ on one boundary arc and λ on the complementary arc—the zero level line of h joining the endpoints of these arcs converges to SLE(4) as the domain grows larger. If instead the boundary values are −a < 0 on the first arc and b > 0 on the complementary arc, then the convergence is to SLE(4; a/λ - 1, b/λ - 1), a variant of SLE(4).     

The discrete Gaussian free field on a compact manifold

In this article we define the discrete Gaussian free field (DGFF) on a compact manifold. Since there is no canonical grid approximation of a manifold, we construct a random graph that suitably replaces the square lattice ${\mathbb{Z}}^d$ in Euclidean space, and prove that the scaling limit of the DGFF is given by the manifold continuum Gaussian free field (GFF). Furthermore using Voronoi tessellations we can interpret the DGFF as element of a Sobolev space and show convergence to the GFF in law with respect to the strong Sobolev topology.     

Tightness of the recentered maximum of the two‐dimensional discrete Gaussian free field

We consider the maximum of the discrete two‐dimensional Gaussian free field (GFF) in a box and prove that its maximum, centered at its mean, is tight, settling a longstanding conjecture. The proof combines a recent observation by Bolthausen, Deuschel, and Zeitouni with elements from Bramson's results on branching Brownian motion and comparison theorems for Gaussian fields. An essential part of the argument is the precise evaluation, up to an error of order 1, of the expected value of the maximum of the GFF in a box. Related Gaussian fields, such as the GFF on a two‐dimensional torus, are also discussed. © 2011 Wiley Periodicals, Inc.     

A continuum model for free growth in living materials

We focus in this work on isotropically growing materials. An adaptive algorithm is used in order to maintain a stress-free state during growth if no external loads are applied, but keeping the volume growth defined by a former kinetic. The proposed model is based on a modified multiplicative split of the deformation gradient into a growth part and an elastic part. The growth part will be isotropic if the elastic deformations are favourable, otherwise the growth will find a more comfortable direction. Three-dimensional examples based on different kinetics are presented and discussed using the numerical model. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exponential and double exponential tails for maximum of two-dimensional discrete Gaussian free field

We study the tail behavior for the maximum of discrete Gaussian free field on a 2D box with Dirichlet boundary condition after centering by its expectation. We show that it exhibits an exponential decay for the right tail and a double exponential decay for the left tail. In particular, our result implies that the variance of the maximum is of order 1, improving an $o(\log n)$ bound by Chatterjee (Chaos, concentration, and multiple valleys, 2008) and confirming a folklore conjecture. An important ingredient for our proof is a result of Bramson and Zeitouni (Commun. Pure Appl. Math, 2010), who proved the tightness of the centered maximum together with an evaluation of the expectation up to an additive constant.     

A comparative study of Gaussian geostatistical models and Gaussian Markov random field models

Gaussian geostatistical models (GGMs) and Gaussian Markov random fields (GMRFs) are two distinct approaches commonly used in spatial models for modeling point-referenced and areal data, respectively. In this paper, the relations between GGMs and GMRFs are explored based on approximations of GMRFs by GGMs, and approximations of GGMs by GMRFs. Two new metrics of approximation are proposed : (i) the Kullback-Leibler discrepancy of spectral densities and (ii) the chi-squared distance between spectral densities. The distances between the spectral density functions of GGMs and GMRFs measured by these metrics are minimized to obtain the approximations of GGMs and GMRFs. The proposed methodologies are validated through several empirical studies. We compare the performance of our approach to other methods based on covariance functions, in terms of the average mean squared prediction error and also the computational time. A spatial analysis of a dataset on PM_2.5 collected in California is presented to illustrate the proposed method.     

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 walk (when time and space are appropriately scaled), the GFF is the limit of many incrementally varying random functions on d-dimensional grids. We present an overview of the GFF and some of the properties that are useful in light of recent connections between the GFF and the Schramm–Loewner evolution. Partially supported by NSF grant DMS0403182.     

Data-driven estimation of atomistic support for continuum stress using the Gaussian mixture model

The notion of stress being an inherent continuum concept has been a matter of discussion at the atomistic level. The atomistic stress measure at a given spatial position contains a space averaging volume over nearby atoms to provide an averaged macroscopic stress measure. Previous work on atomistic stress measures introduce the characteristic length as an a priori given parameter. In this contribution we learn the characteristic length directly from the atomistic data itself. Central to our proposed approach is the grouping of atoms with highly similar values of position and stress into the same atomistic sub-population. We hypothesise that atoms with similar values for position and stress are those atoms which harbour the greatest influence over each other and therefore should be contained within the same space averaging volume. Consequently the characteristic length can be computed directly from the discovered sub-populations by averaging over the maximum extent of each sub-population. We motivate the Gaussian mixture model (GMM) as a principled probabilistic method of estimating the similarity between atoms within position-stress space. The GMM parameters are learnt from the atomistic data using the Expectation Maximization (EM) algorithm. To form a parsimonious representation of the dataset we regularise our model using the Bayesian Information Criterion (BIC) which maintains a balance between too few and too many atomistic sub-populations. We use the GMM to segment the atoms into homogeneous sub-populations based on the probability of each atom belonging to a particular sub-population. Thorough evaluation is conducted on a numerical example of an edge dislocation in a single crystal. We derive estimates of the space averaging volume which are in very close agreement to the corresponding analytical solution. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the first Gaussian map for Prym-canonical line bundles

We prove by degeneration to Prym-canonical binary curves that the first Gaussian map $\mu _A$ of the Prym canonical line bundle $\omega _C \otimes A$ is surjective for the general point $[C,A] \in \mathcal{R }_g$ if $g \ge 12$ , while it is injective if $g \le 11$ .     

Discretization error for the maximum of a Gaussian field

The paper considers the difference between (a) the true maximum of a Gaussian field on a square and (b) its maximum on a regular grid. This difference is called the discretization error. A kind of Slepian model is used to study the behavior of the field around the location of the maximum. We show that the normalized discretization error can be bounded by a quantity that converges to a uniform variable, depending on the Hessian matrix at the point of the maximum. The bound is applied to simulated and real data (satellite positioning data).     

Lacunary free actions on the real line

All groups of free homeomorphisms of the real line are determined up to topological conjugacy. Surprisingly, many of them are lacunary in the sense that no orbit is dense, although the groups themselves (with the exception of the infinite cyclic group) are dense subgroups of $\text{R}$₊. Such pathological behaviour is, however, impossible for normal subgroups of transitive groups.     

A countably compact free Abelian group of size continuum that admits a non-trivial convergent sequence

We show that it is consistent with ZFC that the free Abelian group of cardinality c admits a topological group topology that makes it countably compact with a non-trivial convergent sequence.     

Conditional distributions of the maximum of a Gaussian random field

It is shown that the conditional probabilities of the maximum of a Gaussian random field, under the condition of fixing that element of the field on which this maximum is attained, may be degenerate in the infinite-dimensional case.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 260–263, 1990.