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.     

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.     

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.     

Hadamard’s formula and couplings of SLEs with free field

The relation between level lines of Gaussian free fields (GFF) and SLE₄-type curves was discovered by O. Schramm and S. Sheffield. A weak interpretation of this relation is the existence of a coupling of the GFF and a random curve, in which the curve behaves like a level line of the field. In the present paper we study these couplings for the free field with different boundary conditions. We provide a unified way to determine the law of the curve (i.e. to compute the driving process of the Loewner chain) given boundary conditions of the field and to prove existence of the coupling. The proof is reduced to the verification of two simple properties of the mean and covariance of the field, which always relies on Hadamard’s formula and properties of harmonic functions. Examples include combinations of Dirichlet, Neumann and Riemann–Hilbert boundary conditions. In doubly connected domains, the standard annulus SLE₄ is coupled with a compactified GFF obeying Neumann boundary conditions on the inner boundary. We also consider variants of annulus SLE coupled with free fields having other natural boundary conditions. These include boundary conditions leading to curves connecting two points on different boundary components with prescribed winding as well as those recently proposed by C. Hagendorf, M. Bauer and D. Bernard.     

Universal Padé approximants with respect to the chordal metric

The use of the chordal distance does not change the notion of Universal Taylor series. However, it changes the notion of Universal Padé approximants. Using Padé approximants of meromorphic or holomorphic functions we can approximate all rational functions on compact sets of arbitrary connectivity.     

Holomorphic two-spheres in complex Grassmann manifold <Emphasis Type="Italic">G</Emphasis>(2, 4)

In this paper, we use the harmonic sequence to study the linearly full holomorphic two-spheres in complex Grassmann manifold G(2, 4). We show that if the Gaussian curvature K (with respect to the induced metric) of a non-degenerate holomorphic two-sphere satisfies K ≤ 2 (or K ≥ 2), then K must be equal to 2. Simultaneously, we show that one class of the holomorphic two-spheres with constant curvature 2 is totally geodesic. Concerning the degenerate holomorphic two-spheres, if its Gaussian curvature K ≤ 1 (or K ≥ 1), then K = 1. Moreover, we prove that all holomorphic two-spheres with constant curvature 1 in G(2, 4) must be U (4)-equivalent.     

Bounding the Clique‐Width of H‐Free Chordal Graphs

A graph is H‐free if it has no induced subgraph isomorphic to H. Brandstädt, Engelfriet, Le, and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique‐width. Brandstädt, Le, and Mosca erroneously claimed that the gem and co‐gem are the only two 1‐vertex P₄‐extensions H for which the class of H‐free chordal graphs has bounded clique‐width. In fact we prove that bull‐free chordal and co‐chair‐free chordal graphs have clique‐width at most 3 and 4, respectively. In particular, we find four new classes of H‐free chordal graphs of bounded clique‐width. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions we determine all graphs H for which the class of H‐free chordal graphs has bounded clique‐width. We illustrate the usefulness of this classification for classifying other types of graph classes by proving that the class of ‐free graphs has bounded clique‐width via a reduction to K₄‐free chordal graphs. Finally, we give a complete classification of the (un)boundedness of clique‐width of H‐free weakly chordal graphs.     

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).     

Several results on chordal bipartite graphs

The question of generalizing results involving chordal graphs to similar concepts for chordal bipartite graphs is addressed. First, it is found that the removal of a bisimplicial edge from a chordal bipartite graph produces a chordal bipartite graph. As consequence, occurance of arithmetic zeros will not terminate perfect Gaussian elimination on sparse matrices having associated a chordal bipartite graph. Next, a property concerning minimal edge separators is presented. Finally, it is shown that, to any vertex of a chordal bipartite graph an edge may be added such that the chordality is maintained.     

A class of non-Gaussian second order random fields

Non-Gaussian stochastic fields are introduced by means of integrals with respect to independently scattered stochastic measures distributed according to generalized Laplace laws. In particular, we discuss stationary second order random fields that, as opposed to their Gaussian counterpart, have a possibility of accounting for asymmetry and heavier tails. Additionally to this greater flexibility the models discussed continue to share most spectral properties with Gaussian processes. Their statistical distributions at crossing levels are computed numerically via the generalized Rice formula. The potential for stochastic modeling of real life phenomena that deviate from the Gaussian paradigm is exemplified by a stochastic field model with Matérn covariances.     

共形复Finsler流形上的交换公式

设M为n维复流形,M^-～=T~（1,0）M-{0},F为M上的强拟凸复Finsler度量, F^-=e^σF为F的共形变换。本文得到定义在M^-上的各种Hermitian张量场分别关于复Finsler流形（M,F）和（M,F^-）的复Rund联络求共变微分的各种交换公式。     

Quantum stochastic calculus and quantum Gaussian processes

In this lecture we present a brief outline of boson Fock space stochastic calculus based on the creation, conservation and annihilation operators of free field theory, as given in the 1984 paper of Hudson and Parthasarathy [9]. We show how a part of this architecture yields Gaussian fields stationary under a group action. Then we introduce the notion of semigroups of quasifree completely positive maps on the algebra of all bounded operators in the boson Fock space Γ(? ⁿ ) over ? ⁿ . These semigroups are not strongly continuous but their preduals map Gaussian states to Gaussian states. They were first introduced and their generators were shown to be of the Lindblad type by Vanheuverzwijn [19]. They were recently investigated in the context of quantum information theory by Heinosaari et al. [7]. Here we present the exact noisy Schrödinger equation which dilates such a semigroup to a quantum Gaussian Markov process.     

Perfect Elimination and Chordal Bipartite Graphs

We define two types of bipartite graphs, chordal bipartite graphs and perfect elimination bipartite graphs, and prove theorems analogous to those of Dirac and Rose for chordal graphs (rigid circuit graphs, triangulated graphs). Our results are applicable to Gaussian elimination on sparse matrices where a sequence of pivots preserving zeros is sought. Our work removes the constraint imposed by Haskins and Rose that pivots must be along the main diagonal.     

Free Holomorphic Functions on the Regular Polyball

In this paper, we study free holomorphic functions on the regular polyball, which was recently introduced by Popescu (Adv Math 279:104–158, 2015, Trans Am Math Soc 368:4357–4416, 2016). The purpose of this paper is to continue the line of Popescu to develop a theory of free holomorphic functions. Some results from the classical holomorphic function theory have free analogues in this noncommutative setting. In particular, we prove the maximum principle, Weierstrass and Montel type theorems for free holomorphic functions. As an application, we construct a metric on \(\mathrm{Hol}(\mathbf{B_n}(\mathcal {H}))\) such that it becomes a complete space.     

Extremal holomorphic diffusion processes

Extremal holomorphic diffusion processes are studied. We formulate a stochastic control problem for the extremal function of a set. We then characterize the extremal holomorphic diffusion processes as the optimal diffusion processes of the problem. By making use of SDE representation for the processes, we show that they move on an integral submanifold of the coefficients vector fields of the SDE passing through the starting point.     

Nevanlinna theory through the Brownian motion

In this paper, we introduce the Nevanlinna theory using stochastic calculus, following the works of Davis(1975), Carne(1986) and Atsuji(1995, 2005, 2008 and 2017), etc. In particular, we give(another) proofs of the classical result of Nevanlinna for meromorphic functions and the result of Cartan-Ahlfors for holomorphic curves by using the probabilistic method.     

The Loewner driving function of trajectory arcs of quadratic differentials

We obtain a first order differential equation for the driving function of the chordal Loewner differential equation in the case where the domain is slit by a curve which is a trajectory arc of a certain type of quadratic differential. In particular this includes the case when the curve is a path on the square, triangle or hexagonal lattice in the upper half-plane or, indeed, in any domain with boundary on the lattice. We also demonstrate how we use this to calculate the driving function numerically. Equivalent results for other variants of the Loewner differential equation are also obtained.     

Transverse Lie jets and holomorphic geometric objects on transverse bundles

Two holomorphic fields of geometric objects on a transverse Weil bundle are called equivalent if there exists a holomorphic diffeomorphism of this bundle onto itself which induces the identity transformation of the base manifold and maps one of these fields into the other. In terms of transverse Lie jets, we establish necessary and sufficient conditions for a holomorphic field of geometric objects on a transverse Weil bundle to be equivalent to the prolongation of a field of foliated geometric objects given on the base manifold. As an example, we consider a holomorphic linear connection on a transverse bundle.     

Stochastic Integrals and Evolution Equations with Gaussian Random Fields

The paper studies stochastic integration with respect to Gaussian processes and fields. It is more convenient to work with a field than a process: by definition, a field is a collection of stochastic integrals for a class of deterministic integrands. The problem is then to extend the definition to random integrands. An orthogonal decomposition of the chaos space of the random field, combined with the Wick product, leads to the Itô-Skorokhod integral, and provides an efficient tool to study the integral, both analytically and numerically. For a Gaussian process, a natural definition of the integral follows from a canonical correspondence between random processes and a special class of random fields. Also considered are the corresponding linear stochastic evolution equations.