Authors:Alexis Drouot Pages: 357 - 396 Abstract: Abstract Kinetic Brownian motion on the cosphere bundle of a Riemannian manifold \({\mathbb{M}}\) is a stochastic process that models the geodesic equation perturbed by a random white force of size \({\varepsilon}\) . When \({\mathbb{M}}\) is compact and negatively curved, we show that the L 2-spectrum of the infinitesimal generator of this process converges to the Pollicott–Ruelle resonances of \({\mathbb{M}}\) as \({\varepsilon}\) goes to 0. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2956-0 Issue No:Vol. 356, No. 2 (2017)

Authors:Estelle Basor; Pavel Bleher Pages: 397 - 425 Abstract: Abstract We obtain an asymptotic formula, as \({n\to\infty}\) , for the monomer–monomer correlation function \({K_2(n)}\) in the classical dimer model on a triangular lattice, with the horizontal and vertical weights \({w_h=w_v=1}\) and the diagonal weight \({w_d=t > 0}\) , between two monomers at vertices q and r that are n spaces apart in adjacent rows. We find that \({t_c=\frac{1}{2}}\) is a critical value of t. We prove that in the subcritical case, \({0 < t < \frac{1}{2}}\) , as \({n\to\infty, K_2(n)=K_2(\infty)\left[1-\frac{e^{-n/\xi}}{n}\,\Big(C_1+C_2(-1)^n+ \mathcal{O}(n^{-1})\Big) \right]}\) , with explicit formulae for \({K_2(\infty), \xi, C_1}\) , and \({C_2}\) . In the supercritical case, \({\frac{1}{2} < t < 1}\) , we prove that as \({n\to\infty, K_2(n)=K_2(\infty)\Bigg[1-\frac{e^{-n/\xi}}{n}\, \Big(C_1\cos(\omega n+\varphi_1)+C_2(-1)^n\cos(\omega n+\varphi_2)+ C_3+C_4(-1)^n + \mathcal{O}(n^{-1})\Big)\Bigg]}\) , with explicit formulae for \({K_2(\infty), \xi,\omega}\) , and \({C_1, C_2, C_3, C_4, \varphi_1, \varphi_2}\) . The proof is based on an extension of the Borodin–Okounkov–Case–Geronimo formula to block Toeplitz determinants and on an asymptotic analysis of the Fredholm determinants in hand. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2985-8 Issue No:Vol. 356, No. 2 (2017)

Authors:N. Martynchuk; K. Efstathiou Pages: 427 - 449 Abstract: Abstract The notion of fractional monodromy was introduced by Nekhoroshev, Sadovskií and Zhilinskií as a generalization of standard (‘integer’) monodromy in the sense of Duistermaat from torus bundles to singular torus fibrations. In the present paper we prove a general result that allows one to compute fractional monodromy in various integrable Hamiltonian systems. In particular, we show that the non-triviality of fractional monodromy in 2 degrees of freedom systems with a Hamiltonian circle action is related only to the fixed points of the circle action. Our approach is based on the study of a specific notion of parallel transport along Seifert manifolds. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2988-5 Issue No:Vol. 356, No. 2 (2017)

Authors:Sergey Bravyi; David Gosset Pages: 451 - 500 Abstract: Abstract We give a quasi-polynomial time classical algorithm for estimating the ground state energy and for computing low energy states of quantum impurity models. Such models describe a bath of free fermions coupled to a small interacting subsystem called an impurity. The full system consists of n fermionic modes and has a Hamiltonian \({H=H_0+H_{imp}}\) , where H 0 is quadratic in creation–annihilation operators and H imp is an arbitrary Hamiltonian acting on a subset of O(1) modes. We show that the ground energy of H can be approximated with an additive error \({2^{-b}}\) in time \({n^3 \exp{[O(b^3)]}}\) . Our algorithm also finds a low energy state that achieves this approximation. The low energy state is represented as a superposition of \({\exp{[O(b^3)]}}\) fermionic Gaussian states. To arrive at this result we prove several theorems concerning exact ground states of impurity models. In particular, we show that eigenvalues of the ground state covariance matrix decay exponentially with the exponent depending very mildly on the spectral gap of H 0. A key ingredient of our proof is Zolotarev’s rational approximation to the \({\sqrt{x}}\) function. We anticipate that our algorithms may be used in hybrid quantum-classical simulations of strongly correlated materials based on dynamical mean field theory. We implemented a simplified practical version of our algorithm and benchmarked it using the single impurity Anderson model. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2976-9 Issue No:Vol. 356, No. 2 (2017)

Authors:Matthew D. Blair; Christopher D. Sogge Pages: 501 - 533 Abstract: Abstract We prove a Kakeya–Nikodym bound on eigenfunctions and quasimodes, which sharpens a result of the authors (Blair and Sogge in Anal PDE 8:747–764, 2015) and extends it to higher dimensions. As in the prior work, the key intermediate step is to prove a microlocal version of these estimates, which involves a phase space decomposition of these modes that is essentially invariant under the bicharacteristic/geodesic flow. In a companion paper (Blair and Sogge in J Differ Geom, 2015), it will be seen that these sharpened estimates yield improved L q (M) bounds on eigenfunctions in the presence of nonpositive curvature when \({2 < q < \frac{2(d+1)}{d-1}}\) . PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2977-8 Issue No:Vol. 356, No. 2 (2017)

Authors:Nicolai Reshetikhin; Ananth Sridhar Pages: 535 - 565 Abstract: Abstract The main result of this paper is the construction of infinitely many conserved quantities (corresponding to commuting transfer-matrices) for the limit shape equation for the six vertex model on a cylinder. This suggests that the limit shape equation is an integrable PDE with gradient constraints. At the free fermionic point this equation becomes the complex Burgers equation. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2983-x Issue No:Vol. 356, No. 2 (2017)

Authors:Philip Candelas; Xenia de la Ossa; Jock McOrist Pages: 567 - 612 Abstract: Abstract Heterotic vacua of string theory are realised, at large radius, by a compact threefold with vanishing first Chern class together with a choice of stable holomorphic vector bundle. These form a wide class of potentially realistic four-dimensional vacua of string theory. Despite all their phenomenological promise, there is little understanding of the metric on the moduli space of these. What is sought is the analogue of special geometry for these vacua. The metric on the moduli space is important in phenomenology as it normalises D-terms and Yukawa couplings. It is also of interest in mathematics, since it generalises the metric, first found by Kobayashi, on the space of gauge field connections, to a more general context. Here we construct this metric, correct to first order in \({\alpha^{\backprime}}\) , in two ways: first by postulating a metric that is invariant under background gauge transformations of the gauge field, and also by dimensionally reducing heterotic supergravity. These methods agree and the resulting metric is Kähler, as is required by supersymmetry. Checking the metric is Kähler is intricate and the anomaly cancellation equation for the H field plays an essential role. The Kähler potential nevertheless takes a remarkably simple form: it is the Kähler potential of special geometry with the Kähler form replaced by the \({\alpha^{\backprime}}\) -corrected hermitian form. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2978-7 Issue No:Vol. 356, No. 2 (2017)

Authors:Takuro Mochizuki; Masaki Yoshino Pages: 613 - 625 Abstract: Abstract We study singular monopoles on open subsets in the 3-dimensional Euclidean space. We give two characterizations of Dirac type singularities. One is given in terms of the growth order of the norms of sections which are invariant by the scattering map. The other is given in terms of the growth order of the norms of the Higgs fields. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2981-z Issue No:Vol. 356, No. 2 (2017)

Authors:Indranil Biswas; Viktoria Heu; Jacques Hurtubise Pages: 627 - 640 Abstract: Abstract For the universal isomonodromic deformation of an irreducible logarithmic rank two connection over a smooth complex projective curve of genus at least two, consider the family of holomorphic vector bundles over curves underlying this universal deformation. In a previous work we proved that the vector bundle corresponding to a general parameter of this family is stable. Here we prove that the vector bundle corresponding to a general parameter is in fact very stable, meaning it does not admit any nonzero nilpotent Higgs field. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2987-6 Issue No:Vol. 356, No. 2 (2017)

Authors:Farzad Fathizadeh; Matilde Marcolli Pages: 641 - 671 Abstract: Abstract We show that, when considering the scaling factor as an affine variable, the coefficients of the asymptotic expansion of the spectral action on a (Euclidean) Robertson–Walker spacetime are periods of mixed Tate motives, involving relative motives of complements of unions of hyperplanes and quadric hypersurfaces and divisors given by unions of coordinate hyperplanes. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2991-x Issue No:Vol. 356, No. 2 (2017)

Authors:Valter Pohjola; Leo Tzou Pages: 107 - 142 Abstract: Abstract We prove a fixed frequency inverse scattering result for the magnetic Schrödinger operator (or connection Laplacian) on surfaces with Euclidean ends. We show that, under suitable decaying conditions, the scattering matrix for the operator determines both the gauge class of the connection and the zeroth order potential. PubDate: 2017-11-01 DOI: 10.1007/s00220-017-2982-y Issue No:Vol. 356, No. 1 (2017)

Authors:Maximilian Jeblick; David Mitrouskas; Sören Petrat; Peter Pickl Pages: 143 - 187 Abstract: Abstract The dynamics of a particle coupled to a dense and homogeneous ideal Fermi gas in two spatial dimensions is studied. We analyze the model for coupling parameter g = 1 (i.e., not in the weak coupling regime), and prove closeness of the time evolution to an effective dynamics for large densities of the gas and for long time scales of the order of some power of the density. The effective dynamics is generated by the free Hamiltonian with a large but constant energy shift which is given at leading order by the spatially homogeneous mean field potential of the gas particles. Here, the mean field approximation turns out to be accurate although the fluctuations of the potential around its mean value can be arbitrarily large. Our result is in contrast to a dense bosonic gas in which the free motion of a tracer particle would be disturbed already on a very short time scale. The proof is based on the use of strong phase cancellations in the deviations of the microscopic dynamics from the mean field time evolution. PubDate: 2017-11-01 DOI: 10.1007/s00220-017-2970-2 Issue No:Vol. 356, No. 1 (2017)

Authors:Roland Bauerschmidt; Paul Bourgade; Miika Nikula; Horng-Tzer Yau Pages: 189 - 230 Abstract: Abstract We study the classical two-dimensional one-component plasma of N positively charged point particles, interacting via the Coulomb potential and confined by an external potential. For the specific inverse temperature \({\beta=1}\) (in our normalization), the charges are the eigenvalues of random normal matrices, and the model is exactly solvable as a determinantal point process. For any positive temperature, using a multiscale scheme of iterated mean-field bounds, we prove that the equilibrium measure provides the local particle density down to the optimal scale of \({N^{o(1)}}\) particles. Using this result and the loop equation, we further prove that the particle configurations are rigid, in the sense that the fluctuations of smooth linear statistics on any scale are \({N^{o(1)}}\) . PubDate: 2017-11-01 DOI: 10.1007/s00220-017-2932-8 Issue No:Vol. 356, No. 1 (2017)

Authors:Luigi Amedeo Bianchi; Francesco Morandin Pages: 231 - 260 Abstract: Abstract We study a generalization of the original tree-indexed dyadic model by Katz and Pavlović for the turbulent energy cascade of the three-dimensional Euler equation. We allow the coefficients to vary with some restrictions, thus giving the model a realistic spatial intermittency. By introducing a forcing term on the first component, the fixed point of the dynamics is well defined and some explicit computations allow us to prove the rich multifractal structure of the solution. In particular the exponent of the structure function is concave in accordance with other theoretical and experimental models. Moreover, anomalous energy dissipation happens in a fractal set of dimension strictly less than 3. PubDate: 2017-11-01 DOI: 10.1007/s00220-017-2974-y Issue No:Vol. 356, No. 1 (2017)

Authors:Juhan Aru; Yichao Huang; Xin Sun Pages: 261 - 283 Abstract: Abstract 2D Liouville quantum gravity (LQG) is used as a toy model for 4D quantum gravity and is the theory of world-sheet in string theory. Recently there has been growing interest in studying LQG in the realm of probability theory: David et al. (Liouville quantum gravity on the Riemann sphere. Commun Math Phys 342(3):869–907, 2016) and Duplantier et al. (Liouville quantum gravity as a mating of trees. ArXiv e-prints: arXiv:1409.7055, 2014) both provide a probabilistic perspective of the LQG on the 2D sphere. In particular, in each of them one may find a definition of the so-called unit area quantum sphere. We examine these two perspectives and prove their equivalence by showing that the respective unit area quantum spheres are the same. This is done by considering a unified limiting procedure for defining both objects. PubDate: 2017-11-01 DOI: 10.1007/s00220-017-2979-6 Issue No:Vol. 356, No. 1 (2017)

Authors:Thomas Moser; Robert Seiringer Pages: 329 - 355 Abstract: Abstract We prove that a system of N fermions interacting with an additional particle via point interactions is stable if the ratio of the mass of the additional particle to the one of the fermions is larger than some critical m*. The value of m* is independent of N and turns out to be less than 1. This fact has important implications for the stability of the unitary Fermi gas. We also characterize the domain of the Hamiltonian of this model, and establish the validity of the Tan relations for all wave functions in the domain. PubDate: 2017-11-01 DOI: 10.1007/s00220-017-2980-0 Issue No:Vol. 356, No. 1 (2017)

Authors:Benjamin Landon; Horng-Tzer Yau Pages: 949 - 1000 Abstract: Abstract We analyze the rate of convergence of the local statistics of Dyson Brownian motion to the GOE/GUE for short times t = o(1) with deterministic initial data V. Our main result states that if the density of states of V is bounded both above and away from 0 down to scales \({\ell \ll t}\) in a small interval of size \({G \gg t}\) around an energy \({E_0}\) , then the local statistics coincide with the GOE/GUE near the energy \({E_0}\) after time t. Our methods are partly based on the idea of coupling two Dyson Brownian motions from Bourgade et al. (Commun Pure Appl Math, 2016), the parabolic regularity result of Erdős and Yau (J Eur Math Soc 17(8):1927–2036, 2015), and the eigenvalue rigidity results of Lee and Schnelli (J Math Phys 54(10):103504, 2013). PubDate: 2017-11-01 DOI: 10.1007/s00220-017-2955-1 Issue No:Vol. 355, No. 3 (2017)

Authors:Trevor Clark; Sebastian van Strien; Sofia Trejo Pages: 1001 - 1119 Abstract: Abstract In this paper we prove complex bounds, also referred to as a priori bounds for C 3, and, in particular, for analytic maps of the interval. Any C 3 mapping of the interval has an asymptotically holomorphic extension to a neighbourhood of the interval. We associate to such a map, a complex box mapping, which provides a kind of Markov structure for the dynamics. Moreover, we prove universal geometric bounds on the shape of the domains and on the moduli between components of the range and domain. Such bounds show that the first return maps to these domains are well controlled, and consequently such bounds form one of the corner stones in many recent results in one-dimensional dynamics, for example: renormalization theory, rigidity, density of hyperbolicity, and local connectivity of Julia sets. PubDate: 2017-11-01 DOI: 10.1007/s00220-017-2958-y Issue No:Vol. 355, No. 3 (2017)

Authors:Wolf-Patrick Düll Pages: 1189 - 1207 Abstract: Abstract We consider a nonlinear Klein–Gordon equation with a quasilinear quadratic term. The Nonlinear Schrödinger (NLS) equation can be derived as a formal approximation equation describing the evolution of the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the quasilinear Klein–Gordon equation. It is the purpose of this paper to present a method that allows one to prove error estimates in Sobolev norms between exact solutions of the quasilinear Klein–Gordon equation and the formal approximation obtained via the NLS equation. The paper contains the first validity proof of the NLS approximation of a nonlinear hyperbolic equation with a quasilinear quadratic term by error estimates in Sobolev spaces. We expect that the method developed in the present paper will allow an answer to the relevant question of the validity of the NLS approximation for other quasilinear hyperbolic systems. PubDate: 2017-11-01 DOI: 10.1007/s00220-017-2966-y Issue No:Vol. 355, No. 3 (2017)

Authors:Jens Mund; Erichardson T. de Oliveira Pages: 1243 - 1282 Abstract: Abstract It is well-known that a (point-localized) free quantum field for massive particles with spin s acting in a Hilbert space has at best scaling dimension s + 1, which excludes its use in the perturbative construction of renormalizable interacting models for higher spin ( \({s\geq 1}\) ). Up to date, such models have been constructed only in the context of gauge theory, at the cost of introducing additional unphysical (ghost) fields and an unphysical (indefinite metric) state space. The unphysical degrees of freedom are divided out by requiring gauge (or BRST) invariance. We construct free quantum fields for higher spin particles that have the same good UV behaviour as the scalar field (scaling dimension one), and at the same time act on a Hilbert space without ghosts. They are localized on semi-infinite strings extending to space-like infinity, but are linearly related to their point-local counterparts. We argue that this is sufficient locality for a perturbative construction of interacting models of the gauge theory type, with a string-independent S-matrix and point-localized interacting observable fields. The usual principle of gauge-invariance is here replaced by the (deeper) principle of locality. PubDate: 2017-11-01 DOI: 10.1007/s00220-017-2968-9 Issue No:Vol. 355, No. 3 (2017)