Fractional Gaussian fields are scalar-valued random functions or generalized functions on an n-dimensional manifold M, indexed by a parameter s. They include white noise (s = 0), Brownian motion (s = 1, n = 1), the 2D Gaussian free field (s = 1, n = 2) and the membrane model (s = 2). These simple objects are ubiquitous in math and science, and can be used as a starting point for constructing non-Gaussian theories. They are sometimes parameterized by the Hurst parameter.
The differential form analogs of these objects are equally natural: for example, instead of considering an instance h(x) of the GFF on ℝ2, one might write h1(x)dx1 + h2(x)dx2 where h1 and h2 are independent GFF instances. This “Gaussian-free-field-based 1-form” can be projected onto curl-free and divergence-free components, which in turn arise as gradients and dual-gradients of independent membrane models.
In general, given k ∈ {0,1,…, n}, an instance of the fractional Gaussian k-form with parameter s ∈ ℝ (abbreviated FGFsk(M)) is given by , where Wk is a k-form-valued white noise. We write
The 1-form FGF11(M) and its gauge-fixed projection FGF11(M)d*=0 arise as low-temperature/small-scale limits of U(1) Yang–Mills gauge theory. When U(1) is replaced with another compact Lie group, the corresponding limit is a Lie-algebra-valued analog of FGF11(M)d*=0, which can be interpreted as a random connection whose curvature is (a Lie-algebra valued analog of) FGF20(M)d=0.
Wilson loop observables of this connection are defined for sufficiently regular “big loops” (trajectories in the space of divergence-free 1-forms obtainable as limits of finite-length loops) in a gauge invariant way, even in the non-abelian case. We define “big surfaces” (trajectories in the space of 2-forms obtainable as limits of smooth surfaces with boundary) and note that Stokes’ theorem converts big-loop integrals of FGF11(M)d*=0 into big-surface integrals of FGF20(M)d=0 or FGF20(M). A type of exponential correlation decay and area law applies within the slabs ℝ2 × [0,1]m but not within ℝn for n > 2.
One may interpret FGF11(M)d*=0 as a random divergence-free vector field, which is conjectured to be the fine-mesh scaling limit of the n-dimensional dimer model when n > 2. (Kenyon proved this for n = 2.) We formulate several conjectures and open problems about scaling limits, including possible off-critical/non-Gaussian limits, whose construction in the Yang–Mills setting is a famous open problem.
We introduce path-conjoined graphs defined for two rooted graphs by joining their roots with a path, and investigate the chromatic symmetric functions of its two generalizations: spider-conjoined graphs and chain-conjoined graphs. By using the composition method developed by Zhou and the third author recently, we obtain neat positive eI-expansions for the chromatic symmetric functions of clique-path-cycle graphs, path-clique-path graphs, and clique-clique-path graphs. We pose the e-positivity conjecture for hat-chains.
The purpose of this paper is to introduce and study λ-infinitesimal BiHom-bialgebras (abbr. λ-infBH-bialgebra) and some related structures. They can be seen as an extension of λ-infinitesimal bialgebras considered by Ebrahimi-Fard, including Joni–Rota’s infinitesimal bialgebras as well as Loday–Ronco’s infinitesimal bialgebras, and including also infinitesimal BiHom-bialgebras introduced by Liu, Makhlouf, Menini, Panaite. In this paper, we provide various relevant constructions and new concepts. Two ways are provided for a unitary (resp. counitary) algebra (resp. coalgebra) to be a λ-infBH-bialgebra and the notion of λ-infBH-Hopf module is introduced and discussed. It is proved, in connection with nonhomogeneous (co)associative BiHom-Yang-Baxter equation, that every (left BiHom-)module (resp., comodule) over an (anti-)quasitriangular (resp., (anti-)coquasitriangular) λ-infBH-bialgebra carries a structure of λ-infBH-Hopf module. Moreover, two approaches to construct BiHom-pre-Lie (co)algebras from λ-infBH-bialgebras are presented.
In this paper, we define the spectrum of an abelian category with respect to a class of wide subcategories. We introduce the notion of support of wide subcategories, and then give a classification of radical wide subcategories of an abelian category using it. We also introduce the notion of prime wide subcategories and give a homeomorphism from a topological space to the prime spectrum space using a notion of classifying support data.
We show that for any 1 ≤ s ≤ 2, there is a periodic continuous function f whose Fourier series is divergent at some point, and whose graph satisfies
Based on the three-ball inequality and the doubling inequality established in [Math. Ann., 2012, 353(4): 1157–1181], we quantify the strong unique continuation established by Koch and Tataru [Comm. Pure Appl. Math., 2001, 54(3): 339–360] for elliptic operators with unbounded lower-order coefficients. We also obtain a quantitative strong unique continuation for eigenfunctions, which we use to prove that two Dirichlet–Laplace–Beltrami operators are gauge equivalent when their corresponding metrics coincide in the neighborhood of the boundary and their boundary spectral data coincide on a subset of positive measure.
The geometric structure of three-dimensional manifolds is a fundamental problem in geometric topology. In this paper, we investigate the ideal circle pattern problem on a surface with boundaries under a cellular decomposition. Employing the variational method, we establish a series of equivalent conditions for the existence of a circle pattern in both Euclidean and hyperbolic background geometries.