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 Annales Henri Poincaré   [SJR: 1.016]   [H-I: 28]   [3 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1424-0637 - ISSN (Online) 1424-0661    Published by Springer-Verlag  [2302 journals]
• Inverse-Closed Algebras of Integral Operators on Locally Compact Groups
• Abstract: Abstract We construct some inverse-closed algebras of bounded integral operators with operator-valued kernels, acting in spaces of vector-valued functions on locally compact groups. To this end we make use of covariance algebras associated to C*-dynamical systems defined by the C*-algebras of right uniformly continuous functions with respect to the left regular representation.
PubDate: 2015-05-01

• Hawking Effect for a Toy Model of Interacting Fermions
• Abstract: Abstract We consider a toy model of interacting Dirac fermions in a 1 + 1 dimensional space time describing the exterior of a star collapsing to a black hole. In this situation, we give a rigorous proof of the Hawking effect, namely that under the associated quantum evolution, an initial vacuum state will converge when t → + ∞ to a thermal state at Hawking temperature. We establish this result both for observables falling into the blackhole along null characteristics and for static observables. We also consider the case of an interaction localized near the star boundary, obtaining similar results. We hence extend to an interacting model previous results of Bachelot and Melnyk, obtained for free Dirac fields.
PubDate: 2015-05-01

• On the Form Factors of Local Operators in the Bazhanov–Stroganov and
Chiral Potts Models
• 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: 2015-05-01

• Zero Modes of Quantum Graph Laplacians and an Index Theorem
• Abstract: Abstract We study zero modes of Laplacians on compact and non-compact metric graphs with general self-adjoint vertex conditions. In the first part of the paper, the number of zero modes is expressed in terms of the trace of a unitary matrix $${\mathfrak{S}}$$ that encodes the vertex conditions imposed on functions in the domain of the Laplacian. In the second part, a Dirac operator is defined whose square is related to the Laplacian. To accommodate Laplacians with negative eigenvalues, it is necessary to define the Dirac operator on a suitable Kreĭn space. We demonstrate that an arbitrary, self-adjoint quantum graph Laplacian admits a factorisation into momentum-like operators in a Kreĭn-space setting. As a consequence, we establish an index theorem for the associated Dirac operator and prove that the zero-mode contribution in the trace formula for the Laplacian can be expressed in terms of the index of the Dirac operator.
PubDate: 2015-05-01

• Erratum to: Inverse-Closed Algebras of Integral Operators on Locally
Compact Groups
• PubDate: 2015-05-01

• Next-to-Leading Order in the Large N Expansion of the Multi-Orientable
Random Tensor Model
• Abstract: Abstract In this paper we analyze in detail the next-to-leading order (NLO) of the recently obtained large N expansion for the multi-orientable (MO) tensor model. From a combinatorial point of view, we find the class of Feynman tensor graphs contributing to this order in the expansion. Each such NLO graph is characterized by the property that it contains a certain non-orientable ribbon subgraph (a non-orientable jacket). Furthermore, we find the radius of convergence and the susceptibility exponent of the NLO series for this model. These results represent a first step towards the larger goal of defining an appropriate double-scaling limit for the MO tensor model.
PubDate: 2015-05-01

• Initial Data Sets with Ends of Cylindrical Type: I. The Lichnerowicz
Equation
• Abstract: Abstract We construct large classes of vacuum general relativistic initial data sets, possibly with a cosmological constant $${\Lambda \in \mathbb{R}}$$ , containing ends of cylindrical type.
PubDate: 2015-05-01

• Structural Stability of a Dynamical System Near a Non-Hyperbolic Fixed
Point
• Abstract: Abstract We prove structural stability under perturbations for a class of discrete-time dynamical systems near a non-hyperbolic fixed point. We reformulate the stability problem in terms of the well-posedness of an infinite-dimensional nonlinear ordinary differential equation in a Banach space of carefully weighted sequences. Using this, we prove existence and regularity of flows of the dynamical system which obey mixed initial and final boundary conditions. The class of dynamical systems we study, and the boundary conditions we impose, arise in a renormalization group analysis of the 4-dimensional weakly self-avoiding walk and the 4-dimensional n-component φ 4 spin model.
PubDate: 2015-04-01

• The Dynamics of a Class of Quasi-Periodic Schrödinger Cocycles
• Abstract: Abstract Let $${f:\mathbb{T} \to\mathbb{R}}$$ be a Morse function of class C 2 with exactly two critical points, let $${\omega \in \mathbb{T}}$$ be Diophantine, and let λ > 0 be sufficiently large (depending on f and ω). For any value of the parameter $${E\in \mathbb{R}}$$ , we make a careful analysis of the dynamics of the skew-product map $$\Phi_E(\theta,r)=\left(\theta+\omega, {\rm \lambda} f(\theta)-E-1/r\right),$$ acting on the “torus” $${\mathbb{T} \times \widehat{\mathbb{R}}}$$ . Here, $${\widehat{\mathbb{R}}}$$ denotes the projective space $${\mathbb{R} \cup\{\infty\}}$$ . The map Φ E is intimately related to the quasi-periodic Schrödinger cocycle $${(\omega, A_E): \mathbb{T}\times \mathbb{R}^2 \to \mathbb{T}\times \mathbb{R}^2,\, (\theta,x)\mapsto (\theta+\omega, A_E(\theta)\cdot x)}$$ , where $${A_E:\mathbb{T}\to {\rm SL}(2,\mathbb{R})}$$ is given by $$A_{E}(\theta)=\left( \begin{array}{ll}0 \quad \quad \quad 1\\ -1 \quad {\rm \lambda} f(\theta)-E \\\end{array}\right),\quad E \in \mathbb{R}.$$ More precisely, (ω, A E ) naturally acts on the space $${\mathbb{T} \times \widehat{\mathbb{R}}}$$ , and Φ E is the map thus obtained. The cocycle (ω, A E ) arises when investigating the eigenvalue equation H θ u = Eu, where H θ is the quasi-periodic Schrödinger operator $$(H_\theta u)_n=-(u_{n+1}+u_{n-1}) + {\rm \lambda} f(\theta+(n-1)\omega)u_n,$$ acting on the space $${l^2(\mathbb{Z})}$$ . It is well known that the spectrum of $${H_\theta,\, \sigma(H)}$$ , is independent of the phase $${\theta \in \mathbb{T}}$$ . Under our assumptions on f, ω and λ, Sinai (in J Stat Phys 46(5–6):861–909, 1987) has shown that σ(H) is a Cantor set, and the operator H θ has a pure-point spectrum, with exponentially decaying eigenfunctions, for a.e. $${\theta \in \mathbb{T}}$$ The analysis of Φ E allows us to derive three main results: The (maximal) Lyapunov exponent of the Schrödinger cocycle (ω, A E ) is $${\gtrsim {\rm log} {\rm \lambda}}$$ , uniformly in
PubDate: 2015-04-01

• Approximation of the Integrated Density of States on Sofic Groups
• Abstract: Abstract In this paper, we study spectral properties of self-adjoint operators on a large class of geometries given via sofic groups. We prove that the associated integrated densities of states can be approximated via finite volume analogues. This is investigated in the deterministic as well as in the random setting. In both cases, we cover a wide range of operators including in particular unbounded ones. The large generality of our setting allows one to treat applications from long-range percolation and the Anderson model. Our results apply to operators on $${\mathbb{Z}^d}$$ , amenable groups, residually finite groups and therefore in particular to operators on trees. All convergence results are established without an ergodic theorem at hand.
PubDate: 2015-04-01

• An Extensive Adiabatic Invariant for the Klein–Gordon Model in the
Thermodynamic Limit
• Abstract: Abstract We construct an extensive adiabatic invariant for a Klein–Gordon chain in the thermodynamic limit. In particular, given a fixed and sufficiently small value of the coupling constant a, the evolution of the adiabatic invariant is controlled up to time scaling as β 1/a for any large enough value of the inverse temperature β. The time scale becomes a stretched exponential if the coupling constant is allowed to vanish jointly with the specific energy. The adiabatic invariance is exhibited by showing that the variance along the dynamics, i.e. calculated with respect to time averages, is much smaller than the corresponding variance over the whole phase space, i.e. calculated with the Gibbs measure, for a set of initial data of large measure. All the perturbative constructions and the subsequent estimates are consistent with the extensive nature of the system.
PubDate: 2015-04-01

• Resurgent Transseries and the Holomorphic Anomaly
• Abstract: Abstract The gauge theoretic large N expansion yields an asymptotic series which requires a nonperturbative completion to be well defined. Recently, within the context of random matrix models, it was shown how to build resurgent transseries solutions encoding the full nonperturbative information beyond the ’t Hooft genus expansion. On the other hand, via large N duality, random matrix models may be holographically described by B-model closed topological strings in local Calabi–Yau geometries. This raises the question of constructing the corresponding holographically dual resurgent transseries, tantamount to nonperturbative topological string theory. This paper addresses this point by showing how to construct resurgent transseries solutions to the holomorphic anomaly equations. These solutions are built upon (generalized) multi-instanton sectors, where the instanton actions are holomorphic. The asymptotic expansions around the multi-instanton sectors have both holomorphic and anti-holomorphic dependence, may allow for resonance, and their structure is completely fixed by the holomorphic anomaly equations in terms of specific polynomials multiplied by exponential factors and up to the holomorphic ambiguities—which generalizes the known perturbative structure to the full transseries. In particular, the anti-holomorphic dependence has a somewhat universal character. Furthermore, in the non-perturbative sectors, holomorphic ambiguities may be fixed at conifold points. This construction shows the nonperturbative integrability of the holomorphic anomaly equations and sets the ground to start addressing large-order analysis and resurgent nonperturbative completions within closed topological string theory.
PubDate: 2015-03-29

• On the Hausdorff Dimension of Newhouse Phenomena
• Abstract: Abstract We show that at the vicinity of a generic dissipative homoclinic unfolding of a surface diffeomorphism, the Hausdorff dimension of the set of parameters for which the diffeomorphism admits infinitely many periodic sinks is at least 1/2.
PubDate: 2015-03-21

• On the Essential Spectrum of Two-Dimensional Pauli Operators with
Repulsive Potentials
• Abstract: Abstract We investigate the spectrum of the two-dimensional Pauli operator, describing a spin- $${\frac{1}{2}}$$ particle in a magnetic field B, with a negative scalar potential V, such that V grows at infinity. In particular, we obtain criteria for discrete and dense pure-point spectrum.
PubDate: 2015-03-21

• Eisenstein Quasimodes and QUE
• Abstract: Abstract We consider the question of quantum unique ergodicity (QUE) for quasimodes on surfaces of constant negative curvature, and conjecture the order of quasimodes that should satisfy QUE. We then show that this conjecture holds for Eisenstein series on $${SL(2,\mathbb{Z})\backslash\mathbb{H}}$$ , extending results of Luo–Sarnak and Jakobson. Moreover, we observe that the equidistribution results of Luo–Sarnak and Jakobson extend to quasimodes of much weaker order—for which QUE is known to fail on compact surfaces—though in this scenario the total mass of the limit measures will decrease. We interpret this stronger equidistribution property in the context of arithmetic QUE, in light of recent joint work with Lindenstrauss (Invent Math 198(1), 219–259, 2014) on joint quasimodes.
PubDate: 2015-03-19

• Optimal Lower Bound of the Resonance Widths for the Helmholtz Resonator
• Abstract: Abstract Under a geometric assumption on the region near the end of its neck, we prove an optimal exponential lower bound on the widths of resonances for a general two-dimensional Helmholtz resonator. An extension of the result to the n-dimensional case, $${n \leq 12}$$ , is also obtained.
PubDate: 2015-03-17

• Dynamical Locality of the Free Maxwell Field
• Abstract: Abstract The extent to which the non-interacting and source-free Maxwell field obeys the condition of dynamical locality is determined in various formulations. Starting from contractible globally hyperbolic spacetimes, we extend the classical field theory to globally hyperbolic spacetimes of arbitrary topology in two ways, obtaining a ‘universal’ theory and a ‘reduced’ theory of the classical free Maxwell field and their corresponding quantisations. We show that the classical and the quantised universal theory fail local covariance and dynamical locality owing to the possibility of having non-trivial radicals in the classical pre-symplectic spaces and non-trivial centres in the quantised *-algebras. The classical and the quantised reduced theory are both locally covariant and dynamically local, thus closing a gap in the discussion of dynamical locality and providing new examples relevant to the question of how theories should be formulated so as to describe the same physics in all spacetimes.
PubDate: 2015-03-06

• The Altshuler–Shklovskii Formulas for Random Band Matrices II: The
General Case
• 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

• Deformations of Axially Symmetric Initial Data and the Mass-Angular
Momentum Inequality
• 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

• Multi-Species Mean Field Spin Glasses. Rigorous Results
• 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

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