Authors:Roland Bauerschmidt; Gordon Slade; Alexandre Tomberg; Benjamin C. Wallace Pages: 375 - 402 Abstract: Abstract We study the four-dimensional n-component \({ \varphi ^4}\) spin model for all integers \({n \ge 1}\) and the four-dimensional continuous-time weakly self-avoiding walk which corresponds exactly to the case \({n=0}\) interpreted as a supersymmetric spin model. For these models, we analyse the correlation length of order p, and prove the existence of a logarithmic correction to mean-field scaling, with power \({\frac 12\frac{n+2}{n+8}}\) , for all \({n \ge 0}\) and \({p > 0}\) . The proof is based on an improvement of a rigorous renormalisation group method developed previously. PubDate: 2017-02-01 DOI: 10.1007/s00023-016-0499-0 Issue No:Vol. 18, No. 2 (2017)

Authors:Federico Camia; Marcin Lis Pages: 403 - 433 Abstract: Abstract We introduce and study a Markov field on the edges of a graph \(\mathcal {G}\) in dimension \(d\ge 2\) whose configurations are spin networks. The field arises naturally as the edge-occupation field of a Poissonian model (a soup) of non-backtracking loops and walks characterized by a spatial Markov property such that, conditionally on the value of the edge-occupation field on a boundary set that splits the graph into two parts, the distributions of the loops and arcs contained in the two parts are independent of each other. The field has a Gibbs distribution with a Hamiltonian given by a sum of terms which involve only edges incident on the same vertex. Its free energy density and other quantities can be computed exactly, and their critical behavior analyzed, in any dimension. PubDate: 2017-02-01 DOI: 10.1007/s00023-016-0524-3 Issue No:Vol. 18, No. 2 (2017)

Authors:Martin Vogel Pages: 435 - 517 Abstract: Abstract We consider a non-self-adjoint h-differential model operator \(P_h\) in the semiclassical limit ( \(h\rightarrow 0\) ) subject to small random perturbations. Furthermore, we let the coupling constant \(\delta \) be \(\mathrm {e}^{-\frac{1}{Ch}}\le \delta \ll h^{\kappa }\) for constants \(C,\kappa >0\) suitably large. Let \(\Sigma \) be the closure of the range of the principal symbol. Previous results on the same model by Hager, Bordeaux-Montrieux and Sjöstrand show that if \(\delta \gg \mathrm {e}^{-\frac{1}{Ch}}\) there is, with a probability close to 1, a Weyl law for the eigenvalues in the interior of the of the pseudospectrum up to a distance \(\gg \left( -h\ln {\delta h}\right) ^{\frac{2}{3}}\) to the boundary of \(\Sigma \) . We study the intensity measure of the random point process of eigenvalues and prove an h-asymptotic formula for the average density of eigenvalues. With this we show that there are three distinct regions of different spectral behavior in \(\Sigma \) : the interior of the pseudospectrum is solely governed by a Weyl law, close to its boundary there is a strong spectral accumulation given by a tunneling effect followed by a region where the density decays rapidly. PubDate: 2017-02-01 DOI: 10.1007/s00023-016-0528-z Issue No:Vol. 18, No. 2 (2017)

Authors:Takuro Matsuta; Tohru Koma; Shu Nakamura Pages: 519 - 528 Abstract: Abstract We improve the Lieb–Robinson bound for a wide class of quantum many-body systems with long-range interactions decaying by power law. As an application, we show that the group velocity of information propagation grows by power law in time for such systems, whereas systems with short-range interactions exhibit a finite group velocity as shown by Lieb and Robinson. PubDate: 2017-02-01 DOI: 10.1007/s00023-016-0526-1 Issue No:Vol. 18, No. 2 (2017)

Authors:Alexandre Belin; Christoph A. Keller; Alexander Maloney Pages: 529 - 557 Abstract: Abstract The space of permutation orbifolds is a simple landscape of two-dimensional CFTs, generalizing the well-known symmetric orbifolds. We consider constraints which a permutation orbifold with large central charge must obey in order to be holographically dual to a weakly coupled (but possibly stringy) theory of gravity in AdS. We then construct explicit examples of permutation orbifolds which obey these constraints. In our constructions, the spectrum remains finite at large N, but differs qualitatively from that of symmetric orbifolds. We also discuss under what conditions the correlation functions factorize at large N and thus reduce to those of a generalized free field in AdS. We show that this happens not just for symmetric orbifolds, but also for permutation groups which act “democratically” in a sense which we define. PubDate: 2017-02-01 DOI: 10.1007/s00023-016-0529-y Issue No:Vol. 18, No. 2 (2017)

Authors:Santiago Codesido; Alba Grassi; Marcos Mariño Pages: 559 - 622 Abstract: Abstract Recently, a correspondence has been proposed between spectral theory and topological strings on toric Calabi–Yau manifolds. In this paper, we develop in detail this correspondence for mirror curves of higher genus, which display many new features as compared to the genus one case studied so far. Given a curve of genus g, our quantization scheme leads to g different trace class operators. Their spectral properties are encoded in a generalized spectral determinant, which is an entire function on the Calabi–Yau moduli space. We conjecture an exact expression for this spectral determinant in terms of the standard and refined topological string amplitudes. This conjecture provides a non-perturbative definition of the topological string on these geometries, in which the genus expansion emerges in a suitable ’t Hooft limit of the spectral traces of the operators. In contrast to what happens in quantum integrable systems, our quantization scheme leads to a single quantization condition, which is elegantly encoded by the vanishing of a quantum-deformed theta function on the mirror curve. We illustrate our general theory by analyzing in detail the resolved \({\mathbb C}^3/{\mathbb Z}_5\) orbifold, which is the simplest toric Calabi–Yau manifold with a genus two mirror curve. By applying our conjecture to this example, we find new quantization conditions for quantum mechanical operators, in terms of genus two theta functions, as well as new number-theoretic properties for the periods of this Calabi–Yau. PubDate: 2017-02-01 DOI: 10.1007/s00023-016-0525-2 Issue No:Vol. 18, No. 2 (2017)

Authors:Teng Huang Pages: 623 - 641 Abstract: Abstract We consider an instanton, A, with L 2-bounded curvature F A on the cylindrical manifold \({Z=\mathbf{R} \times M}\) , where M is a closed Riemannian n-manifold, \({n \geq 4}\) . We assume M admits a smooth 3-form P and a smooth 4-form Q satisfy \({dP=4Q}\) and \({d\ast_{M}{Q}=(n-3)\ast_{M}P}\) . Manifolds with these forms include nearly Kähler 6-manifolds and nearly parallel G 2-manifolds in dimension 7. Then we can prove that the instanton must be a flat connection. PubDate: 2017-02-01 DOI: 10.1007/s00023-016-0503-8 Issue No:Vol. 18, No. 2 (2017)

Authors:Piotr T. Chruściel; Romain Gicquaud Pages: 643 - 679 Abstract: Abstract We give an exhaustive description of bifurcations and of the number of solutions of the vacuum Lichnerowicz equation with positive cosmological constant on \({S^1\times S^2}\) with \({U(1)\times SO(3)}\) -invariant seed data. The resulting CMC slicings of Schwarzschild–de Sitter and Nariai are described. PubDate: 2017-02-01 DOI: 10.1007/s00023-016-0501-x Issue No:Vol. 18, No. 2 (2017)

Authors:Håkan Andréasson; David Fajman; Maximilian Thaller Pages: 681 - 705 Abstract: Abstract We prove existence of spherically symmetric, static, self-gravitating photon shells as solutions to the massless Einstein–Vlasov system. The solutions are highly relativistic in the sense that the ratio 2m(r) / r is close to 8 / 9, where m(r) is the Hawking mass and r is the area radius. In 1955 Wheeler constructed, by numerical means, so-called idealized spherically symmetric geons, i.e., solutions of the Einstein–Maxwell equations for which the energy momentum tensor is spherically symmetric on a time average. The structure of these solutions is such that the electromagnetic field is confined to a thin shell for which the ratio 2m / r is close to 8 / 9, i.e., the solutions are highly relativistic photon shells. The solutions presented in this work provide an alternative model for photon shells or idealized spherically symmetric geons. PubDate: 2017-02-01 DOI: 10.1007/s00023-016-0531-4 Issue No:Vol. 18, No. 2 (2017)

Authors:Jeremiah Birrell; Scott Hottovy; Giovanni Volpe; Jan Wehr Pages: 707 - 755 Abstract: Abstract We study damped geodesic motion of a particle of mass m on a Riemannian manifold, in the presence of an external force and noise. Lifting the resulting stochastic differential equation to the orthogonal frame bundle, we prove that, as \({m \to 0}\) , its solutions converge to solutions of a limiting equation which includes a noise-induced drift term. A very special case of the main result presents Brownian motion on the manifold as a limit of inertial systems. PubDate: 2017-02-01 DOI: 10.1007/s00023-016-0508-3 Issue No:Vol. 18, No. 2 (2017)

Authors:Daniela Cadamuro; Yoh Tanimoto Pages: 233 - 279 Abstract: Abstract We construct candidates for observables in wedge-shaped regions for a class of 1 + 1-dimensional integrable quantum field theories with bound states whose S-matrix is diagonal, by extending our previous methods for scalar S-matrices. Examples include the Z(N)-Ising models, the A N−1-affine Toda field theories and some S-matrices with CDD factors. We show that these candidate operators which are associated with elementary particles commute weakly on a dense domain. For the models with two species of particles, we can take a larger domain of weak commutativity and give an argument for the Reeh–Schlieder property. PubDate: 2017-01-01 DOI: 10.1007/s00023-016-0515-4 Issue No:Vol. 18, No. 1 (2017)

Authors:Takahisa Inui Pages: 307 - 343 Abstract: Abstract We consider the focusing nonlinear cubic Klein–Gordon equation in three spatial dimensions (NLKG). By the classical result of Payne and Sattinger (Israel J Math 22(3–4):273–303, 1975), one can distinguish global existence and blow-up for real-valued initial data with energy less than that of the ground state, by a variational characterization. Following the idea of Kenig and Merle (Acta Math 201(2):147–212, 2008), Ibrahim, Masmoudi, and Nakanishi (Anal PDE 4(3):405–460, 2011) proved scattering of real-valued solutions in the region of global existence. We will extend their classification to complex-valued solutions below the ground-state standing waves using the charge (see Theorem 1.1). We apply their method to obtain the scattering result. The difference is that they use only the energy, but we need to control both the energy and the charge. On the other hand, we cannot use their proof, which is based on Payne and Sattinger (Israel J Math 22(3–4):273–303, 1975), to obtain the blow-up result. Therefore, we combine the idea of Holmer and Roudenko (Commun Partial Differ Equ 35(5):878–905, 2010), who proved a similar classification for the nonlinear Schrödinger equation, with the idea of Ohta and Todorova (SIAM J Math Anal 38(6):1912–1931, 2007). Moreover, we extend the classification using not only the energy and the charge, but also the momentum (see Theorem 1.3). Due to these extensions, we can classify the solutions which have large energy. PubDate: 2017-01-01 DOI: 10.1007/s00023-016-0510-9 Issue No:Vol. 18, No. 1 (2017)

Authors:Yannis Angelopoulos; Stefanos Aretakis; Dejan Gajic Abstract: Abstract We show that degenerate horizons exhibit a new trapping effect. Specifically, we obtain a non-degenerate Morawetz estimate for the wave equation in the domain of outer communications of extremal Reissner–Nordström up to and including the future event horizon. We show that such an estimate requires (1) a higher degree of regularity for the initial data, reminiscent of the regularity loss in the high-frequency trapping estimates on the photon sphere, and (2) the vanishing of an explicit quantity that depends on the restriction of the initial data on the horizon. The latter condition demonstrates that degenerate horizons exhibit a new \(L^{2}\) concentration phenomenon (namely, a global trapping effect, in the sense that this effect is not due to individual underlying null geodesics as in the case of the photon sphere). We moreover uncover a new stable higher-order trapping effect; we show that higher-order estimates do not hold regardless of the degree of regularity and the support of the initial data. We connect our findings to the spectrum of the stability operator in the theory of marginally outer trapped surfaces. Our methods and results play a crucial role in our upcoming works on linear and nonlinear wave equations on extremal black hole backgrounds. PubDate: 2017-01-10 DOI: 10.1007/s00023-016-0545-y

Authors:Benoît Laslier; Fabio Lucio Toninelli Abstract: Abstract We study a reversible continuous-time Markov dynamics on lozenge tilings of the plane, introduced by Luby et al. (SIAM J Comput 31:167–192, 2001). Single updates consist in concatenations of n elementary lozenge rotations at adjacent vertices. The dynamics can also be seen as a reversible stochastic interface evolution. When the update rate is chosen proportional to 1 / n, the dynamics is known to have special features: a certain Hamming distance between configurations contracts with time on average (Luby et al. in SIAM J Comput 31:167–192, 2001), and the relaxation time of the Markov chain is diffusive (Wilson in Ann Appl Probab 14:274–325, 2004), growing like the square of the diameter of the system. Here, we present another remarkable feature of this dynamics, namely we derive, in the diffusive timescale, a fully explicit hydrodynamic limit equation for the height function (in the form of a nonlinear parabolic PDE). While this equation cannot be written as a gradient flow w.r.t. a surface energy functional, it has nice analytic properties, for instance it contracts the \({\mathbb {L}}^2\) distance between solutions. The mobility coefficient \(\mu \) in the equation has non-trivial but explicit dependence on the interface slope and, interestingly, is directly related to the system’s surface free energy. The derivation of the hydrodynamic limit is not fully rigorous, in that it relies on an unproven assumption of local equilibrium. PubDate: 2017-01-10 DOI: 10.1007/s00023-016-0548-8

Authors:Benjamin Landon; Annalisa Panati; Jane Panangaden; Justine Zwicker Abstract: Abstract The dynamic reflection probability of Davies and Simon (Commun Math Phys 63(3):277–301, 1978) and the spectral reflection probability of Gesztesy et al. (Diff Integral Eqs 10(3):521–546, 1997) and Gesztesy and Simon (Helv Phys Acta 70:66–71, 1997) for a one-dimensional Schrödinger operator \(H = - \Delta + V\) are characterized in terms of the scattering theory of the pair \((H, H_\infty )\) where \(H_\infty \) is the operator obtained by decoupling the left and right half-lines \(\mathbb {R}_{\le 0}\) and \(\mathbb {R}_{\ge 0}\) . An immediate consequence is that these reflection probabilities are in fact the same, thus providing a short and transparent proof of the main result of Breuer et al. (Commun Math Phys 295(2):531–550, 2010). This approach is inspired by recent developments in non-equilibrium statistical mechanics of the electronic black-box model and follows a strategy parallel to Jakšić (Commun Math Phys 332:827–838, 2014). PubDate: 2017-01-09 DOI: 10.1007/s00023-016-0543-0

Authors:Sven Bachmann; Martin Fraas; Gian Michele Graf Abstract: Abstract We study the evolution of a driven harmonic oscillator with a time-dependent frequency \(\omega _t \propto t \) . At time \(t=0\) , the Hamiltonian undergoes a point of infinite spectral degeneracy. If the system is initialized in the instantaneous vacuum in the distant past, then the asymptotic future state is a squeezed state, whose parameters are explicitly determined. We show that the squeezing is independent on the sweeping rate. This manifests the failure of the adiabatic approximation at points, where infinitely many eigenvalues collide. We extend our analysis to the situation, where the gap at \(t=0\) remains finite. We also discuss the natural geometry of the manifold of squeezed states. We show that it is realized by the Poincaré disk model viewed as a Kähler manifold. PubDate: 2017-01-05 DOI: 10.1007/s00023-016-0539-9

Authors:Edgar Gasperín; Juan A. Valiente Kroon Abstract: Abstract The conformal structure of the Schwarzschild–de Sitter spacetime is analysed using the extended conformal Einstein field equations. To this end, initial data for an asymptotic initial value problem for the Schwarzschild–de Sitter spacetime are obtained. This initial data allow to understand the singular behaviour of the conformal structure at the asymptotic points where the horizons of the Schwarzschild–de Sitter spacetime meet the conformal boundary. Using the insights gained from the analysis of the Schwarzschild–de Sitter spacetime in a conformal Gaussian gauge, we consider nonlinear perturbations close to the Schwarzschild–de Sitter spacetime in the asymptotic region. We show that small enough perturbations of asymptotic initial data for the Schwarzschild–de Sitter spacetime give rise to a solution to the Einstein field equations which exists to the future and has an asymptotic structure similar to that of the Schwarzschild–de Sitter spacetime. PubDate: 2017-01-04 DOI: 10.1007/s00023-016-0544-z

Authors:Per von Soosten; Simone Warzel Abstract: Abstract We apply Feshbach–Krein–Schur renormalization techniques in the hierarchical Anderson model to establish a criterion on the single-site distribution which ensures exponential dynamical localization as well as positive inverse participation ratios and Poisson statistics of eigenvalues. Our criterion applies to all cases of exponentially decaying hierarchical hopping strengths and holds even for spectral dimension \( d > 2 \) , which corresponds to the regime of transience of the underlying hierarchical random walk. This challenges recent numerical findings that the spectral dimension is significant as far as the Anderson transition is concerned. PubDate: 2017-01-03 DOI: 10.1007/s00023-016-0549-7

Authors:Aghil Alaee; Marcus Khuri; Hari Kunduri Abstract: Abstract We prove a mass-angular momentum-charge inequality for a broad class of maximal, asymptotically flat, bi-axisymmetric initial data within the context of five-dimensional minimal supergravity. We further show that the charged Myers–Perry black hole initial data are the unique minimizers. Also, we establish a rigidity statement for the relevant BPS bound, and give a variational characterization of BMPV black holes. PubDate: 2017-01-03 DOI: 10.1007/s00023-016-0542-1

Authors:Chris Bourne; Johannes Kellendonk; Adam Rennie Abstract: Abstract We study the application of Kasparov theory to topological insulator systems and the bulk–edge correspondence. We consider observable algebras as modelled by crossed products, where bulk and edge systems may be linked by a short exact sequence. We construct unbounded Kasparov modules encoding the dynamics of the crossed product. We then link bulk and edge Kasparov modules using the Kasparov product. Because of the anti-linear symmetries that occur in topological insulator models, real \(C^*\) -algebras and KKO-theory must be used. PubDate: 2017-01-03 DOI: 10.1007/s00023-016-0541-2