We prove that the two-dimensional Schroedinger operator with a potential having the symmetry of a honeycomb structure has dispersion surfaces with conical singularities (Dirac points) at the vertices of its Brillouin zone. No assumptions are made on the size of the potential. We then prove the robustness of such conical singularities to a restrictive class of perturbations, which break the honeycomb lattice symmetry. General small perturbations of potentials with Dirac points do not have Dirac points; their dispersion surfaces are smooth. The presence of Dirac points in honeycomb structures is associated with many novel electronic and optical properties of materials such as graphene.

/pdf/honeycomb-lattice-potentials-and-dirac-points-3uun3jkp22.pdf

Honeycomb lattice potentials and Dirac points

We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovasz local lemma in probabilistic combinatorics. We show that the conclusion of the Lovasz local lemma holds for dependency graph G and probabilities {p
 x
 } if and only if the independent-set polynomial for G is nonvanishing in the polydisc of radii {p
 x
 }. Furthermore, we show that the usual proof of the Lovasz local lemma – which provides a sufficient condition for this to occur – corresponds to a simple inductive argument for the nonvanishing of the independent-set polynomial in a polydisc, which was discovered implicitly by Shearer(98) and explicitly by Dobrushin.(37,38) We also present some refinements and extensions of both arguments, including a generalization of the Lovasz local lemma that allows for ‘‘soft’’ dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternating-sign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.

/pdf/the-repulsive-lattice-gas-the-independent-set-polynomial-and-13ze0hayzj.pdf

The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lovasz Local Lemma

* Preface * Introduction Part I: Compactifications of Riemannian Symmetric Spaces * Review of Classical Compactifications of Symmetric Spaces * Uniform Construction of Compactifications of Symmetric Spaces * Properties of Compactifications of Symmetric Spaces Part II: Smooth Compactifications of Semisimple Symmetric Spaces * Smooth Compactifications of Riemannian Symmetric Spaces G / K * Semisimple Symmetric Spaces G / H * The Real Points of Complex Symmetric Spaces Defined Over R * The DeConcini-Procesi Compactification of a Complex Symmetric Space and its Real Points * The Oshima-Sekiguchi Compactification of G / K and Comparison with G/Hw (R) Part III: Compactifications of Locally Symmetric Spaces * Classical Compactifications of Locally Symmetric Spaces * Uniform Construction of Compactifications of Locally Symmetric Spaces * Properties of Compactifications of Locally Symmetric Spaces * Subgroup Compactifications of o \ G * Metric Properties of Compactifications of Locally Symmetric Spaces o \ X * References * Index

http://dept.math.lsa.umich.edu/~lji/head.pdf

Compactifications of Symmetric and Locally Symmetric Spaces

Asymptotics of the principal components estimator of large factor models with weakly influential factors

C onstructing a distance that is invariant under a given class of mappings is one of the fundamental tools for the geometric approach in mathematics. The idea goes back to Klein and even to Riemann. In this article we will consider distances invariant under biholomorphic mappings of complex manifolds. There will be many such distances. A number of these will come from functions on the tangent spaces, in the way that a Riemannian metric on a manifold yields a distance on the manifold. Following Riemann’s lead, we think of a suitable function on the tangent spaces as a way to measure “lengths” of tangent vectors, and we can use it to define lengths of curves and ultimately a distance on the manifold. We do not insist that this function be related to an inner product. If the complex manifold is M and its tangent spaces are denoted Tp(M), then we shall work with any nonnegative function f(p, v) , for v ∈ Tp(M) , such that f(p, v) suitably respects scalar multiplication (real or complex as appropriate) in v . In most cases the function will be continuous in (p, v), but sometimes we allow the continuity to be slightly relaxed. Motivated by the Riemannian case, we shall refer to this function as a metric if f(p, v) vanishes only for v = 0, or a pseudometric in general. We do not assume that f(p, v) satisfies the triangle inequality in v if p is fixed; if the triangle inequality is in fact satisfied, the function will be called a norm or pseudonorm. If the “length” function comes from an inner product, then the metric is said to be Riemannian as usual. If that inner product is the real part of a Hermitian inner product, then we call the metric Hermitian. If the Hermitian inner product behaves almost like the Euclidean metric (in the sense that they agree to order two), then we call the metric Kählerian. If, on the other hand, there is no inner product present at all and only the notion of vector length is defined, then we call the metric Finslerian. The way that the word “Hermitian” is used here may be unfamiliar to some readers, and we offer a note of clarification by considering the simple example of Cn. If we think of Cn as a complex space, then it is natural to use the inner product, for z = (z1, . . . , zn) and w = (w1, . . . , wn) in Cn, given by 〈z,w〉 = n ∑

/pdf/invariant-distances-and-metrics-in-complex-analysis-43c42jl7wt.pdf

Invariant distances and metrics in complex analysis

Fundamental theory: Holomorphic functions and domains of holomorphy Implicit functions and analytic sets The Poincare, Cousin, and Runge problems Pseudoconvex domains and pseudoconcave sets Holomorphic mappings Theory of analytic spaces: Ramified domains Analytic sets and holomorphic functions Analytic spaces Normal pseudoconvex spaces Bibliography Index.

博史山口

Papers

Function theory in several complex variables

博史 山口

Papers

Function theory in several complex variables

博史山口