Abstract: Abstract
We study magnetic quantum Hall systems in a half-plane with Dirichlet boundary conditions along the edge. Much work has been done on the analysis of the currents associated with states whose energy is located between Landau levels. These edge states carry a non-zero current that remains well-localized in a neighborhood of the boundary. In this article, we study the behavior of states with energies close to a Landau level. Such states are referred to as bulk states in the physics literature. Since the magnetic Schrödinger operator is invariant with respect to translations along the edge, it is a direct integral of operators indexed by a real wave number. We analyse these fiber operators and prove new asymptotics on the band functions and their first derivative as the wave number goes to infinity. We apply these results to prove that the current carried by a bulk state is small compared to the current carried by an edge state. We also prove that the bulk states are small near the edge. PubDate: 2014-12-09

Abstract: Abstract
We consider families of random non-unitary contraction operators defined as deformations of CMV matrices which appear naturally in the study of random quantum walks on trees or lattices. We establish several deterministic and almost sure results about the location and nature of the spectrum of such non-normal operators as a function of their parameters. We relate these results to the analysis of certain random quantum walks, the dynamics of which can be studied by means of iterates of such random non-unitary contraction operators. PubDate: 2014-12-07

Abstract: Abstract
We explore the possibility that the Higgs boson of the standard model be actually a member of a larger family, by showing that a more elaborate internal structure naturally arises from geometrical arguments, in the context of a partly original handling of gauge fields which was put forward in previous papers. A possible mechanism yielding the usual Higgs potential is proposed. New types of point interactions, arising in particular from two-spinor index contractions, are shown to be allowed. PubDate: 2014-12-07

Abstract: Abstract
The Kasner metrics are among the simplest solutions of the vacuum Einstein equations, and we use them here to examine the conformal method of finding solutions of the Einstein constraint equations. After describing the conformal method’s construction of constant mean curvature (CMC) slices of Kasner spacetimes, we turn our attention to non-CMC slices of the smaller family of flat Kasner spacetimes. In this restricted setting we obtain a full description of the construction of certain U
n-1 symmetric slices, even in the far-from-CMC regime. Among the conformal data sets generating these slices we find that most data sets construct a single flat Kasner spacetime, but that there are also far-from-CMC data sets that construct one-parameter families of slices. Although these non-CMC families are analogues of well-known CMC one-parameter families, they differ in important ways. Most significantly, unlike the CMC case, the condition signaling the appearance of these non-CMC families is not naturally detected from the conformal data set itself. In light of this difficulty, we propose modifications of the conformal method that involve a conformally transforming mean curvature. PubDate: 2014-12-06

Abstract: Abstract
We discuss a one-dimensional, periodic charge density wave model, and a toy model introduced in Kaspar and Mungan (EPL 103:46002, 2013) which is intended to approximate the former in the case of strong pinning. For both systems an external force may be applied, driving it in one of two (±) directions, and we describe an avalanche algorithm producing an ordered sequence of static configurations leading to the depinning thresholds. For the toy model these threshold configurations are explicit functions of the underlying quenched disorder, as is the threshold-to-threshold evolution via iteration of the algorithm. These explicit descriptions are used to study the law of the random polarization P for the toy model in two cases. Evolving from a macroscopically flat initial state to threshold, we give a scaling limit characterization determines the final value of P. Evolving from (−)-threshold to (+)-threshold, we use an identification with record sequences to study P as a function of the difference between the current force F and the threshold force F
th. The results presented are rigorous and give strong evidence that the depinning transition in the toy model is a dynamic critical phenomenon. PubDate: 2014-12-05

Abstract: Abstract
The aim of this paper is twofold. First, we study eigenvalues and eigenvectors of the adjacency matrix of a bond percolation graph when the base graph is finite and well approximated locally by an infinite regular graph. We relate quantitatively the empirical measure of the eigenvalues and the delocalization of the eigenvectors to the spectrum of the adjacency operator of the percolation on the infinite graph. Secondly, we prove that percolation on an infinite regular tree with degree at least three preserves the existence of an absolutely continuous spectrum if the removal probability is small enough. These two results are notably relevant for bond percolation on a uniformly sampled regular graph or a Cayley graph with large girth. PubDate: 2014-12-05

Abstract: Abstract
This paper is concerned with computing the spectral dimension of (critical) 2d-Liouville quantum gravity. As a warm-up, we first treat the simple case of boundary Liouville quantum gravity. We prove that the spectral dimension is 1 via an exact expression for the boundary Liouville Brownian motion and heat kernel. Then we treat the 2d-case via a decomposition of time integral transforms of the Liouville heat kernel into Gaussian multiplicative chaos of Brownian bridges. We show that the spectral dimension is 2 in this case, as derived by physicists (see Ambjørn et al. in JHEP 9802:010, 1998) 15 years ago. PubDate: 2014-12-01

Abstract: Abstract
Let
\({\Omega \subset \mathbb{R}^2}\)
be an open, bounded domain and
\({\Omega = \bigcup_{i = 1}^{N} \Omega_{i}}\)
be a partition. Denote the Fraenkel asymmetry by
\({0 \leq \mathcal{A}(\Omega_i) \leq 2}\)
and write
$$D(\Omega_i) := \frac{ \Omega_{i} - {\rm min}_{1 \leq j \leq N}{ \Omega_{j} }}{ \Omega_{i} }$$
with
\({0 \leq D(\Omega_{i}) \leq 1}\)
. For N sufficiently large depending only on
\({\Omega}\)
, there is an uncertainty principle
$$\left(\sum_{i=1}^{N}{\frac{ \Omega_{i} }{ \Omega }{\mathcal{A}}(\Omega_i)}\right) + \left(\sum_{i=1}^{N}{\frac{ \Omega_i }{ \Omega }D(\Omega_i)}\right) \geq \frac{1}{60000}.$$
The statement remains true in dimensions
\({n \geq 3}\)
for some constant
\({c_{n} > 0}\)
. As an application, we give an (unspecified) improvement of Pleijel’s estimate on the number of nodal domains of a Laplacian eigenfunction and an improved inequality for a spectral partition problem. PubDate: 2014-12-01

Abstract: Abstract
We show absence of positive eigenvalues for generalized N-body hard-core Schrödinger operators under the condition of bounded obstacles with connected exterior. A particular example is atoms and molecules with the assumption of infinite mass and finite extent nuclei. PubDate: 2014-12-01

Abstract: Abstract
We consider homogeneous Bose gas in a large cubic box with periodic boundary conditions, at zero temperature. We analyze its excitation spectrum in a certain kind of a mean-field infinite-volume limit. We prove that under appropriate conditions the excitation spectrum has the form predicted by the Bogoliubov approximation. Our result can be viewed as an extension of the result of Seiringer (Commun. Math. Phys. 306:565–578, 2011) to large volumes. PubDate: 2014-12-01

Abstract: Abstract
This paper deals with two fundamental models for convection in a reacting fluid and porous medium with magnetic field effect. We demonstrate that the solution depends continuously on changes in the chemical reaction and the electrical conductivity coefficients. The continuous dependence is unconditional in two dimensions but conditional in three dimensions. PubDate: 2014-12-01

Abstract: Abstract
In this paper, we investigate the existence of solutions for a system of BPS vortex equations arising from the theory of multiple intersecting D-branes. Using a direct minimization method, we establish a sharp existence and uniqueness theorem for this system over a doubly periodic domain and over the full plane, respectively. In particular, we obtain an explicit necessary and sufficient condition, explicitly expressed in terms of the vortex numbers and the size of the domain, for the existence of a solution of the system in the doubly periodic domain case. PubDate: 2014-12-01

Abstract: Abstract
We study the spectrum of a one-dimensional Dirac operator pencil, with a coupling constant in front of the potential considered as the spectral parameter. Motivated by recent investigations of graphene waveguides, we focus on the values of the coupling constant for which the kernel of the Dirac operator contains a non-trivial square integrable function. In physics literature such a function is called a confined zero mode. Several results on the asymptotic distribution of coupling constants giving rise to zero modes are obtained. In particular, we show that this distribution depends in a subtle way on the sign variation and the presence of gaps in the potential. Surprisingly, it also depends on the arithmetic properties of certain quantities determined by the potential. We further observe that variable sign potentials may produce complex eigenvalues of the operator pencil. Some examples and numerical calculations illustrating these phenomena are presented. PubDate: 2014-12-01

Abstract: Abstract
This is an expository account of Balaban’s approach to the renormalization group. The method is illustrated with a treatment of the ultraviolet problem for the scalar
\({\phi^4}\)
model on a toroidal lattice in dimension d = 3. In this third paper we demonstrate convergence of the expansion and complete the proof of a stability bound. PubDate: 2014-11-01

Abstract: Abstract
The vector space
\({\otimes^{n}\mathbb{C}^2}\)
upon which the XXZ Hamiltonian with n spins acts bears the structure of a module over both the Temperley–Lieb algebra
\({{\rm TL}_{n}(\beta = q + q^{-1})}\)
and the quantum algebra
\({{\rm U}_{q} \mathfrak{sl}_2}\)
. The decomposition of
\({\otimes^{n}\mathbb{C}^2}\)
as a
\({{\rm U}_{q} \mathfrak{sl}_2}\)
-module was first described by Rosso (Commun Math Phys 117:581–593, 1988), Lusztig (Cont Math 82:58–77, 1989) and Pasquier and Saleur (Nucl Phys B 330:523–556, 1990) and that as a TL
n
-module by Martin (Int J Mod Phys A 7:645–673, 1992) (see also Read and Saleur Nucl Phys B 777(3):316–351, 2007; Gainutdinov and Vasseur Nucl Phys B 868:223–270, 2013). For q generic, i.e. not a root of unity, the TL
n
-module
\({\otimes^{n}\mathbb{C}^2}\)
is known to be a sum of irreducible modules. We construct the projectors (idempotents of the algebra of endomorphisms of
\({\otimes^{n}\mathbb{C}^2}\)
) onto each of these irreducible modules as linear combinations of elements of
\({{\rm U}_{q} \mathfrak{sl}_2}\)
. When q = q
c
is a root of unity, the TL
n
-module
\({\otimes^{n}\mathbb{C}^2}\)
(with n large enough) can be written as a direct sum of indecomposable modules that are not all irreducible. We also give the idempotents projecting onto these indecomposable modules. Their expression now involves some new generators, whose action on
\({\otimes^{n}\mathbb{C}^2}\)
is that of the divided powers
\({(S^{\pm})^{(r)} = \lim_{q \rightarrow q_{c}} (S^{\pm})^r/[r]!}\)
. PubDate: 2014-11-01

Abstract: Abstract
Melonic graphs constitute the family of graphs arising at leading order in the 1/N expansion of tensor models. They were shown to lead to a continuum phase, reminiscent of branched polymers. We show here that they are in fact precisely branched polymers, that is, they possess Hausdorff dimension 2 and spectral dimension 4/3. PubDate: 2014-11-01

Abstract: Abstract
In the early 1980s Yau posed the problem of establishing the rigidity of the Hawking–Penrose singularity theorems. Approaches to this problem have involved the introduction of Lorentzian Busemann functions and the study of the geometry of their level sets—the horospheres. The regularity theory in the Lorentzian case is considerably more complicated and less complete than in the Riemannian case. In this paper, we introduce a broad generalization of the notion of horosphere in Lorentzian geometry and take a completely different (and highly geometric) approach to regularity. These generalized horospheres are defined in terms of achronal limits, and the improved regularity we obtain is based on regularity properties of achronal boundaries. We establish a splitting result for generalized horospheres, which when specialized to Cauchy horospheres yields new results on the Bartnik splitting conjecture, a concrete realization of the problem posed by Yau. Our methods are also applied to spacetimes with positive cosmological constant. We obtain a rigid singularity result for future asymptotically de Sitter spacetimes related to results in Andersson and Galloway (Adv Theor Math Phys 6:307–327, 2002), and Cai and Galloway (Adv Theor Math Phys 3:1769–1783, 2000). PubDate: 2014-11-01

Abstract: In this paper, we present a straightforward pictorial representation of the double affine Hecke algebra (DAHA) which enables us to translate the abstract algebraic structure of a DAHA into an intuitive graphical calculus suitable for a physics audience. Initially, we define the larger double affine Q-dependent braid group. This group is constructed by appending to the braid group a set of operators Q
i
, before extending it to an affine Q-dependent braid group. We show specifically that the elliptic braid group and the DAHA can be obtained as quotient groups. Complementing this, we present a pictorial representation of the double affine Q-dependent braid group based on ribbons living in a toroid. We show that in this pictorial representation, we can fully describe any DAHA. Specifically, we graphically describe the parameter q upon which this algebra is dependent and show that in this particular representation q corresponds to a twist in the ribbon. PubDate: 2014-11-01

Abstract: Abstract
In this paper, we reformulate the combinatorial core of constructive quantum field theory. We define universal rational combinatorial weights for pairs made of a graph and any of its spanning trees. These weights are simply the percentage of Hepp’s sectors of the graph in which the tree is leading, in the sense of Kruskal’s greedy algorithm. Our main new mathematical result is an integral representation of these weights in terms of the positive matrix appearing in the symmetric “BKAR” Taylor forest formula. Then, we explain how the new constructive technique called Loop Vertex Expansion reshuffles according to these weights the divergent series of the intermediate field representation into a convergent series which is the Borel sum of the ordinary perturbative Feynman’s series. PubDate: 2014-11-01