Abstract: Abstract
The Altshuler–Shklovskii formulas (Altshuler and Shklovskii, BZh Eksp Teor Fiz 91:200, 1986) predict, for any disordered quantum system in the diffusive regime, a universal power law behaviour for the correlation functions of the mesoscopic eigenvalue density. In this paper and its companion (Erdős and Knowles, The Altshuler–Shklovskii formulas for random band matrices I: the unimodular case, 2013), we prove these formulas for random band matrices. In (Erdős and Knowles, The Altshuler–Shklovskii formulas for random band matrices I: the unimodular case, 2013) we introduced a diagrammatic approach and presented robust estimates on general diagrams under certain simplifying assumptions. In this paper, we remove these assumptions by giving a general estimate of the subleading diagrams. We also give a precise analysis of the leading diagrams which give rise to the Altschuler–Shklovskii power laws. Moreover, we introduce a family of general random band matrices which interpolates between real symmetric (β = 1) and complex Hermitian (β = 2) models, and track the transition for the mesoscopic density–density correlation. Finally, we address the higher-order correlation functions by proving that they behave asymptotically according to a Gaussian process whose covariance is given by the Altshuler–Shklovskii formulas. PubDate: 2015-03-01

Abstract: Abstract
We describe neutrino radiation in general relativity by introducing the energy–momentum tensor of a null fluid into the Einstein equations. Investigating the geometry and analysis at null infinity, we prove that a component of the null fluid enlarges the Christodoulou memory effect of gravitational-waves. The description of neutrinos in general relativity as a null fluid can be regarded as a limiting case of a more general description using the massless limit of the Einstein–Vlasov system. Gigantic neutrino bursts occur in our universe in core-collapse supernovae and in the mergers of neutron star binaries. PubDate: 2015-03-01

Abstract: Abstract
We show how to reduce the general formulation of the mass-angular momentum inequality, for axisymmetric initial data of the Einstein equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. This procedure is based on a certain deformation of the initial data which preserves the relevant geometry, while achieving the maximal condition and its implied inequality (in a weak sense) for the scalar curvature; this answers a question posed by R. Schoen. The primary equation involved, bears a strong resemblance to the Jang-type equations studied in the context of the positive mass theorem and the Penrose inequality. Each equation in the system is analyzed in detail individually, and it is shown that appropriate existence/uniqueness results hold with the solution satisfying desired asymptotics. Lastly, it is 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 and angular momentum. PubDate: 2015-03-01

Abstract: Abstract
We study a multi-species spin glass system where the density of each species is kept fixed at increasing volumes. The model reduces to the Sherrington–Kirkpatrick one for the single species case. The existence of the thermodynamic limit is proved for all density values under a convexity condition on the interaction. The thermodynamic properties of the model are investigated and the annealed, the replica-symmetric and the replica symmetry breaking bounds are proved using Guerra’s scheme. The annealed approximation is proved to be exact under a high-temperature condition. We show that the replica-symmetric solution has negative entropy at low temperatures. We study the properties of a suitably defined replica symmetry breaking solution and we optimize it within a novel ziggurat ansatz. The generalized order parameter is described by a Parisi-like partial differential equation. PubDate: 2015-03-01

Abstract: Abstract
We study the analogues of irreducibility, period, and communicating classes for open quantum random walks, as defined in (J Stat Phys 147(4):832–852, 2012). We recover results similar to the standard ones for Markov chains, in terms of ergodic behaviour, decomposition into irreducible subsystems, and characterization of invariant states. PubDate: 2015-02-19

Abstract: Abstract
In 1969, Choquet-Bruhat and Geroch established the existence of a unique maximal globally hyperbolic Cauchy development of given initial data for the Einstein equations. Their proof, however, has the unsatisfactory feature that it relies crucially on the axiom of choice in the form of Zorn’s lemma. In this paper, we present a proof that avoids the use of Zorn’s lemma. In particular, we provide an explicit construction of this maximal globally hyperbolic development. PubDate: 2015-02-15

Abstract: Abstract
We propose a way to study one-dimensional statistical mechanics models with complex-valued action using transfer operators. The argument consists of two steps. First, the contour of integration is deformed so that the associated transfer operator is a perturbation of a normal one. Then the transfer operator is studied using methods of semi-classical analysis. In this paper, we concentrate on the second step, the main technical result being a semi-classical estimate for powers of an integral operator which is approximately normal. PubDate: 2015-02-15

Abstract: In this work, we present an abstract framework that allows to obtain mixing (and in some cases sharp mixing) rates for a reasonable large class of invertible systems preserving an infinite measure. The examples explicitly considered are the invertible analogue of both Markov and non-Markov unit interval maps. For these examples, in addition to optimal results on mixing and rates of mixing in the infinite case, we obtain results on the decay of correlation in the finite case of invertible non-Markov maps, which, to our knowledge, were not previously addressed. The proposed method consists of a combination of the framework of operator renewal theory, as introduced in the context of dynamical systems by Sarig (Invent Math 150:629–653, 2002), with the framework of function spaces of distributions developed in the recent years along the lines of Blank et al. (Nonlinearity 15:1905–1973, 2001). PubDate: 2015-02-14

Abstract: Abstract
We exam the validity of the definition of the ADM angular momentum without the parity assumption. Explicit examples of asymptotically flat hypersurfaces in the Minkowski spacetime with zero ADM energy–momentum vector and finite non-zero angular momentum vector are presented. We also discuss the Beig–Ó Murchadha–Regge–Teitelboim center of mass and study analogous examples in the Schwarzschild spacetime. PubDate: 2015-02-11

Abstract: Abstract
We extend a randomization method, introduced by Shiffman–Zelditch and developed by Burq–Lebeau on compact manifolds for the Laplace operator, to the case of
\({\mathbb{R}^d}\)
with the harmonic oscillator. We construct measures, thanks to probability laws which satisfies the concentration of measure property, on the support of which we prove optimal-weighted Sobolev estimates on
\({\mathbb{R}^d}\)
. This construction relies on accurate estimates on the spectral function in a non-compact configuration space. As an application, we show that there exists a basis of Hermite functions with good decay properties in
\({L^{\infty}(\mathbb{R}^{d})}\)
, when d ≥ 2. PubDate: 2015-02-01

Abstract: Abstract
We consider the initial value problem for a three-component system of quadratic derivative nonlinear Schrödinger equations in two space dimensions with the masses satisfying the resonance relation. We present a structural condition on the nonlinearity under which small data global existence holds. It is also shown that the solution is asymptotically free. Our proof is based on the commuting vector field method combined with smoothing effects. PubDate: 2015-02-01

Abstract: Abstract
We consider generalised versions of the spin-boson model at small coupling. We assume the spin (or atom) to sit at the origin
\({0 \in \mathbb{R}^d}\)
and the propagation speed v
p
of free bosons to be constant, i.e. independent of momentum. In particular, the bosons are massless. We prove detailed bounds on the mean number of bosons contained in the ball
\({\{ x \leq v_p t \}}\)
. In particular, we prove that, as
\({t \to \infty}\)
, this number tends to an asymptotic value that can be naturally identified as the mean number of bosons bound to the atom in the ground state. Physically, this means that bosons, that are not bound to the atom, are travelling outwards at a speed that is not lower than v
p
, hence the term ‘minimal velocity estimate’. Additionally, we prove bounds on the number of emitted bosons with low momentum (soft mode bounds). This paper is an extension of our earlier work in De Roeck and Kupiainen (Annales Henri Poincaré 14:253–311, 2013). Together with the results in De Roeck and Kupiainen (Annales Henri Poincaré 14:253–311, 2013), the bounds of the present paper suffice to prove asymptotic completeness, as we describe in De Roeck et al. (Asymptotic completeness in the massless spin-boson model, 2012). PubDate: 2015-02-01

Abstract: Abstract
Our previous constructions of Borchers triples are extended to massless scattering with nontrivial left and right components. A massless Borchers triple is constructed from a set of left–left, right–right and left–right scattering functions. We find a correspondence between massless left–right scattering S-matrices and massive block diagonal S-matrices. We point out a simple class of S-matrices with examples. We study also the restriction of two-dimensional models to the lightray. Several arguments for constructing strictly local two-dimensional nets are presented and possible scenarios are discussed. PubDate: 2015-02-01

Abstract: Abstract
We study a class of random block operators which appear as effective one-particle Hamiltonians for the anisotropic XY quantum spin chain in an exterior magnetic field given by an array of i.i.d. random variables. For arbitrary non-trivial single-site distribution of the magnetic field, we prove dynamical localization of these operators at non-zero energy. PubDate: 2015-02-01

Abstract: Abstract
We show the explicit agreement between the derivation of the Bekenstein–Hawking entropy of a Euclidean BTZ black hole from the point of view of spin foam models and canonical quantization. This is done by considering a graph observable (corresponding to the black hole horizon) in the Turaev–Viro state sum model, and then analytically continuing the resulting partition function to negative values of the cosmological constant. PubDate: 2015-02-01

Abstract: Abstract
Starting from the form factor expansion in finite volume, we derive the multidimensional generalization of the so-called Natte series for the time- and distance-dependent reduced density matrix at zero temperature in the non-linear Schrödinger model. This representation allows one to read-off straightforwardly the long-time/large-distance asymptotic behaviour of this correlator. This method of analysis reduces the complexity of the computation of the asymptotic behaviour of correlation functions in the so-called interacting integrable models, to the one appearing in free-fermion equivalent models. We compute explicitly the first few terms appearing in the asymptotic expansion. Part of these terms stems from excitations lying away from the Fermi boundary, and hence go beyond what can be obtained using the CFT/Luttinger liquid-based predictions. PubDate: 2015-02-01

Abstract: Abstract
In this paper, we describe the weak limits of the measures associated to the eigenfunctions of the Laplacian on a Quantum graph for a generic metric in terms of the Gauss map of the determinant manifold. We describe also all the limits with minimal support (the “scars”). PubDate: 2015-02-01

Abstract: Abstract
We construct a family of transforms labeled by (ν, m) and mapping isometrically square integrable functions on a finite subset of
\({\mathbb{R}}\)
onto L
2-eigenspaces associated with the discrete spectrum of a charged particle evolving in the Riemann sphere under influence of a uniform magnetic field with a strength proportional to
\({2\nu \in \mathbb{Z}_{+}^{\ast}}\)
. These transforms are attached to spherical Landau levels
\({\lambda _{m}^{\nu}=\left( 2m+1\right) \nu +m\left( m+1\right)}\)
with
\({m\in \mathbb{Z}_{+}}\)
and are called discrete Bargmann transforms. PubDate: 2015-02-01

Abstract: Abstract
To any solution of a linear system of differential equations, we associate a matrix kernel, correlators satisfying a set of loop equations, and in the presence of isomonodromic parameters, a Tau function. We then study their semiclassical expansion (WKB type expansion in powers of the weight
\({\hbar}\)
per derivative) of these quantities. When this expansion is of topological type (TT), the coefficients of expansions are computed by the topological recursion with initial data given by the semiclassical spectral curve of the linear system. This provides an efficient algorithm to compute them at least when the semiclassical spectral curve is of genus 0. TT is a non-trivial property, and it is an open problem to find a criterion which guarantees it is satisfied. We prove TT and illustrate our construction for the linear systems associated to the qth reductions of KP—which contain the (p, q) models as a specialization. PubDate: 2015-01-13