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

Abstract: Abstract
We define scattering data for the relativistic Newton equation in a static external electromagnetic field
\({(-\nabla V, B)\in C^1(\mathbb{R}^n,\mathbb{R}^n)\times C^1(\mathbb{R}^n,A_n(\mathbb{R})), n\geq 2}\)
, that decays at infinity like
\({r^{-\alpha-1}}\)
for some
\({\alpha\in (0,1]}\)
, where
\({A_n(\mathbb{R})}\)
is the space of
\({n\times n}\)
antisymmetric matrices. We prove, in particular, that the short-range part of
\({(\nabla V,B)}\)
can be reconstructed from the high-energy asymptotics of the scattering data provided that the long-range tail of
\({(\nabla V,B)}\)
is known. We consider also inverse scattering in other asymptotic regimes. This work generalizes [Jollivet (Asympt Anal 55:103–123, 2007)] where a short-range electromagnetic field was considered. PubDate: 2014-10-28

Abstract: Abstract
We study Jack polynomials in N variables, with parameter α, and having a prescribed symmetry with respect to two disjoint subsets of variables. For instance, these polynomials can exhibit a symmetry of type AS, which means that they are antisymmetric in the first m variables and symmetric in the remaining N − m variables. One of our main goals is to extend recent works on symmetric Jack polynomials (Baratta and Forrester in Nucl Phys B 843:362–381, 2011; Berkesch et al. in Jack polynomials as fractional quantum Hall states and the Betti numbers of the (k + 1)-equals ideal, 2013; Bernevig and Haldane in Phys Rev Lett 101:1–4, 2008) and prove that the Jack polynomials with prescribed symmetry also admit clusters of size k and order r, that is, the polynomials vanish to order r when k + 1 variables coincide. We first prove some general properties for generic α, such as their uniqueness as triangular eigenfunctions of operators of Sutherland type, and the existence of their analogues in infinity many variables. We then turn our attention to the case with α = −(k + 1)/(r − 1). We show that for each triplet (k, r, N), there exist admissibility conditions on the indexing sets, called superpartitions, that guaranty both the regularity and the uniqueness of the polynomials. These conditions are also used to establish similar properties for non-symmetric Jack polynomials. As a result, we prove that the Jack polynomials with arbitrary prescribed symmetry, indexed by (k, r, N)-admissible superpartitions, admit clusters of size k = 1 and order r ≥ 2. In the last part of the article, we find necessary and sufficient conditions for the invariance under translation of the Jack polynomials with prescribed symmetry AS. This allows to find special families of superpartitions that imply the existence of clusters of size k > 1 and order r ≥ 2. PubDate: 2014-10-28

Abstract: Abstract
Given a unitary representation of a Lie group G on a Hilbert space
\({\mathcal H}\)
, we develop the theory of G-invariant self-adjoint extensions of symmetric operators using both von Neumann’s theorem and the theory of quadratic forms. We also analyze the relation between the reduction theory of the unitary representation and the reduction of the G-invariant unbounded operator. We also prove a G-invariant version of the representation theorem for closed and semi-bounded quadratic forms. The previous results are applied to the study of G-invariant self-adjoint extensions of the Laplace–Beltrami operator on a smooth Riemannian manifold with boundary on which the group G acts. These extensions are labeled by admissible unitaries U acting on the L
2-space at the boundary and having spectral gap at −1. It is shown that if the unitary representation V of the symmetry group G is traceable, then the self-adjoint extension of the Laplace–Beltrami operator determined by U is G-invariant if U and V commute at the boundary. Various significant examples are discussed at the end. PubDate: 2014-10-26

Abstract: Abstract
We show how to reduce the general formulation of the mass–angular momentum–charge inequality, for axisymmetric initial data of the Einstein–Maxwell equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. It is also shown that the same reduction argument applies to the basic inequality yielding a lower bound for the area of black holes in terms of mass, angular momentum, and charge. This extends previous work by the authors (Cha and Khuri, Ann Henri Poincaré, doi:10.1007/s00023-014-0332-6, arXiv:1401.3384, 2014), in which the role of charge was omitted. Lastly, we improve upon the hypotheses required for the mass–angular momentum–charge inequality in the maximal case. PubDate: 2014-10-25

Abstract: Abstract
We consider general cyclic representations of the six-vertex Yang–Baxter algebra and analyze the associated quantum integrable systems, the Bazhanov–Stroganov model and the corresponding chiral Potts model on finite size lattices. We first determine the propagator operator in terms of the chiral Potts transfer matrices and we compute the scalar product of separate states (including the transfer matrix eigenstates) as a single determinant formulae in the framework of Sklyanin’s quantum separation of variables. Then, we solve the quantum inverse problem and reconstruct the local operators in terms of the separate variables. We also determine a basis of operators whose form factors are characterized by a single determinant formulae. This implies that the form factors of any local operator are expressed as finite sums of determinants. Among these form factors written in determinant form are in particular those which will reproduce the chiral Potts order parameters in the thermodynamic limit. The results presented here are the generalization to the present models associated to the most general cyclic representations of the six-vertex Yang–Baxter algebra of those we derived for the lattice sine–Gordon model. PubDate: 2014-10-22

Abstract: Abstract
It is shown that solutions to Einstein’s field equations with positive cosmological constant can include non-zero rest-mass fields which coexist with and travel unimpeded across a smooth conformal boundary. This is exemplified by the coupled Einstein-massive-scalar field equations for which the mass m is related to the cosmological constant λ by the relation 3m
2 = 2 λ. Cauchy data for the conformal field equations can in this case be prescribed on the (compact, space-like) conformal boundary
\({\mathcal{J}^{+}}\)
. Their developments backwards in time induce a set of standard Cauchy data on space-like slices for the Einstein-massive-scalar field equations which is open in the set of all Cauchy data for this system. PubDate: 2014-10-18