Authors:Roberto Cortez; Joaquin Fontbona Pages: 913 - 941 Abstract: We prove propagation of chaos at explicit polynomial rates in Wasserstein distance \({\mathcal{W}_{2}}\) for Kac’s N-particle system associated with the spatially homogeneous Boltzmann equation for Maxwell molecules. Our approach is mainly based on novel probabilistic coupling techniques. Combining them with recent stabilization results for the particle system we obtain, under suitable moments assumptions on the initial distribution, a uniform-in-time estimate of order almost \({N^{-1/3}}\) for \({\mathcal{W}_{2}^{2}}\) . PubDate: 2018-02-17 DOI: 10.1007/s00220-018-3101-4 Issue No:Vol. 357, No. 3 (2018)

Authors:Matthias Keller; Yehuda Pinchover; Felix Pogorzelski Abstract: For a given subcritical discrete Schrödinger operator H on a weighted infinite graph X, we construct a Hardy-weight w which is optimal in the following sense. The operator H − λw is subcritical in X for all λ < 1, null-critical in X for λ = 1, and supercritical near any neighborhood of infinity in X for any λ > 1. Our results rely on a criticality theory for Schrödinger operators on general weighted graphs. PubDate: 2018-02-26 DOI: 10.1007/s00220-018-3107-y

Authors:Ovidiu Costin; Rodica D. Costin; Joel L. Lebowitz Abstract: We present a non-perturbative solution of the Schrödinger equation \({i\psi_t(t,x)=-\psi_{xx}(t,x)-2(1 +\alpha \sin\omega t) \delta(x)\psi(t,x)}\) , written in units in which \({\hbar=2m=1}\) , describing the ionization of a model atom by a parametric oscillating potential. This model has been studied extensively by many authors, including us. It has surprisingly many features in common with those observed in the ionization of real atoms and emission by solids, subjected to microwave or laser radiation. Here we use new mathematical methods to go beyond previous investigations and to provide a complete and rigorous analysis of this system. We obtain the Borel-resummed transseries (multi-instanton expansion) valid for all values of α, ω, t for the wave function, ionization probability, and energy distribution of the emitted electrons, the latter not studied previously for this model. We show that for large t and small α the energy distribution has sharp peaks at energies which are multiples of ω, corresponding to photon capture. We obtain small α expansions that converge for all t, unlike those of standard perturbation theory. We expect that our analysis will serve as a basis for treating more realistic systems revealing a form of universality in different emission processes. PubDate: 2018-02-22 DOI: 10.1007/s00220-018-3105-0

Authors:Carlos I. Pérez-Sánchez Abstract: Colored tensor models (CTM) is a random geometrical approach to quantum gravity. We scrutinize the structure of the connected correlation functions of general CTM-interactions and organize them by boundaries of Feynman graphs. For rank-D interactions including, but not restricted to, all melonic \({\varphi^4}\) -vertices—to wit, solely those quartic vertices that can lead to dominant spherical contributions in the large-N expansion—the aforementioned boundary graphs are shown to be precisely all (possibly disconnected) vertex-bipartite regularly edge-D-colored graphs. The concept of CTM-compatible boundary-graph automorphism is introduced and an auxiliary graph calculus is developed. With the aid of these constructs, certain U (∞)-invariance of the path integral measure is fully exploited in order to derive a strong Ward-Takahashi Identity for CTMs with a symmetry-breaking kinetic term. For the rank-3 \({\varphi^4}\) -theory, we get the exact integral-like equation for the 2-point function. Similarly, exact equations for higher multipoint functions can be readily obtained departing from this full Ward-Takahashi identity. Our results hold for some Group Field Theories as well. Altogether, our non-perturbative approach trades some graph theoretical methods for analytical ones. We believe that these tools can be extended to tensorial SYK-models. PubDate: 2018-02-22 DOI: 10.1007/s00220-018-3103-2

Authors:Andrei Neguţ Abstract: We construct the action of the q-deformed W-algebra on its level r representation geometrically, using the moduli space of U(r) instantons on the plane and the double shuffle algebra. We give an explicit LDU decomposition for the action of W-algebra currents in the fixed point basis of the level r representation, and prove a relation between the Carlsson–Okounkov Ext operator and intertwiners for the deformed W-algebra. We interpret this result as a q-deformed version of the AGT–W relations. PubDate: 2018-02-21 DOI: 10.1007/s00220-018-3102-3

Authors:Jason Miller Abstract: Suppose that h is a Gaussian free field (GFF) on a planar domain. Fix \({\kappa \in (0,4)}\) . The \({{\rm SLE}_\kappa}\) light cone L \({(\theta)}\) of h with opening angle \({\theta \in [0,\pi]}\) is the set of points reachable from a given boundary point by angle-varying flow lines of the (formal) vector field \({e^{ih/\chi}}\) , \({\chi = \tfrac{2}{\sqrt{\kappa}} - \tfrac{\sqrt{\kappa}}{2}}\) , with angles in \({[-\tfrac{\theta}{2},\tfrac{\theta}{2}]}\) . We derive the Hausdorff dimension of L \({(\theta)}\) . If \({\theta =0}\) then L \({(\theta)}\) is an ordinary \({{\rm SLE}_{\kappa}}\) curve (with \({\kappa < 4}\) ); if \({\theta = \pi}\) then L \({(\theta)}\) is the range of an \({{\rm SLE}_{\kappa'}}\) curve ( \({\kappa' = 16/\kappa > 4}\) ). In these extremes, this leads to a new proof of the Hausdorff dimension formula for \({{\rm SLE}}\) . We also consider \({{\rm SLE}_\kappa(\rho)}\) processes, which were originally only defined for \({\rho > -\,2}\) , but which can also be defined for \({\rho \leq -2}\) using Lévy compensation. The range of an \({{\rm SLE}_\kappa(\rho)}\) is qualitatively different when \({\rho \leq -2}\) . In particular, these curves are self-intersecting for \({\kappa < 4}\) and double points are dense, while ordinary \({{\rm SLE}_\kappa}\) is simple. It was previously shown (Miller and Sheffield in Gaussian free field light cones and \({{\rm SLE}_\kappa(\rho)}\) , 2016) that certain \({{\rm SLE}_\kappa(\rho)}\) curves agree in law with certain light cones. Combining this with other known results, we obtain a general formula for the Hausdorff dimension of \({{\rm SLE}_\kappa(\rho)}\) for all values of PubDate: 2018-02-20 DOI: 10.1007/s00220-018-3109-9

Authors:Eli Hawkins Abstract: Algebraic quantum field theory is considered from the perspective of the Hochschild cohomology bicomplex. This is a framework for studying deformations and symmetries. Deformation is a possible approach to the fundamental challenge of constructing interacting QFT models. Symmetry is the primary tool for understanding the structure and properties of a QFT model. This perspective leads to a generalization of the algebraic quantum field theory framework, as well as a more general definition of symmetry. This means that some models may have symmetries that were not previously recognized or exploited. To first order, a deformation of a QFT model is described by a Hochschild cohomology class. A deformation could, for example, correspond to adding an interaction term to a Lagrangian. The cohomology class for such an interaction is computed here. However, the result is more general and does not require the undeformed model to be constructed from a Lagrangian. This computation leads to a more concrete version of the construction of perturbative algebraic quantum field theory. PubDate: 2018-02-15 DOI: 10.1007/s00220-018-3098-8

Authors:Ewain Gwynne; Adrien Kassel; Jason Miller; David B. Wilson Abstract: We introduce a two-parameter family of probability measures on spanning trees of a planar map. One of the parameters controls the activity of the spanning tree and the other is a measure of its bending energy. When the bending parameter is 1, we recover the active spanning tree model, which is closely related to the critical Fortuin–Kasteleyn model. A random planar map decorated by a spanning tree sampled from our model can be encoded by means of a generalized version of Sheffield’s hamburger-cheeseburger bijection. Using this encoding, we prove that for a range of parameter values (including the ones corresponding to maps decorated by an active spanning tree), the infinite-volume limit of spanning-tree-decorated planar maps sampled from our model converges in the peanosphere sense, upon rescaling, to an \({{\rm SLE}_\kappa}\) -decorated γ-Liouville quantum cone with \({\kappa > 8}\) and \({\gamma = 4/ \sqrt\kappa \in (0,\sqrt 2)}\) . PubDate: 2018-02-14 DOI: 10.1007/s00220-018-3104-1

Authors:Moritz Doll; Oran Gannot; Jared Wunsch Abstract: Let H denote the harmonic oscillator Hamiltonian on \({\mathbb{R}^d,}\) perturbed by an isotropic pseudodifferential operator of order 1. We consider the Schrödinger propagator \({U(t)=e^{-itH},}\) and find that while \({{\rm sing-supp} {\rm Tr} U(t) \subset 2 \pi \mathbb{Z}}\) as in the unperturbed case, there exists a large class of perturbations in dimensions \({d \geq 2}\) for which the singularities of \({{\rm Tr} U(t)}\) at nonzero multiples of \({2 \pi}\) are weaker than the singularity at t = 0. The remainder term in the Weyl law is of order \({o(\lambda^{d-1})}\) , improving in these cases the \({o(\lambda^{d-1})}\) remainder previously established by Helffer–Robert. PubDate: 2018-02-08 DOI: 10.1007/s00220-018-3100-5

Authors:Duiliu-Emanuel Diaconescu; Ron Donagi; Tony Pantev Abstract: A string theoretic framework is constructed relating the cohomology of wild character varieties to refined stable pair theory and torus link invariants. Explicit conjectural formulas are derived for wild character varieties with a unique irregular point on the projective line. For this case, this leads to a conjectural colored generalization of existing results of Hausel, Mereb and Wong as well as Shende, Treumann and Zaslow. PubDate: 2018-02-08 DOI: 10.1007/s00220-018-3097-9

Authors:Phylippe Eyssidieux; Vincent Guedj; Ahmed Zeriahi Abstract: Studying the behavior of the Kähler–Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge–Ampère equations. In this article, the third of a series on this subject, we study the long term behavior of the normalized Kähler–Ricci flow on mildly singular varieties of positive Kodaira dimension, generalizing results of Song and Tian who dealt with smooth minimal models. PubDate: 2018-02-07 DOI: 10.1007/s00220-018-3087-y

Authors:Clay Córdova; Ben Heidenreich; Alexandr Popolitov; Shamil Shakirov Abstract: We find an exact solution to strongly-coupled matrix models with a single-trace monomial potential. Our solution yields closed form expressions for the partition function as well as averages of Schur functions. The results are fully factorized into a product of terms linear in the rank of the matrix and the parameters of the model. We extend our formulas to include both logarithmic and finite-difference deformations, thereby generalizing the celebrated Selberg and Kadell integrals. We conjecture a formula for correlators of two Schur functions in these models, and explain how our results follow from a general orbifold-like procedure that can be applied to any one-matrix model with a single-trace potential. PubDate: 2018-02-07 DOI: 10.1007/s00220-017-3072-x

Authors:G. Cannizzaro; K. Matetski Abstract: We study a general family of space–time discretizations of the KPZ equation and show that they converge to its solution. The approach we follow makes use of basic elements of the theory of regularity structures (Hairer in Invent Math 198(2):269–504, 2014) as well as its discrete counterpart (Hairer and Matetski in Discretizations of rough stochastic PDEs, 2015. arXiv:1511.06937). Since the discretization is in both space and time and we allow non-standard discretization for the product, the methods mentioned above have to be suitably modified in order to accommodate the structure of the models under study. PubDate: 2018-02-03 DOI: 10.1007/s00220-018-3089-9

Authors:Amin Coja-Oghlan; Charilaos Efthymiou; Nor Jaafari; Mihyun Kang; Tobias Kapetanopoulos Abstract: Diluted mean-field models are spin systems whose geometry of interactions is induced by a sparse random graph or hypergraph. Such models play an eminent role in the statistical mechanics of disordered systems as well as in combinatorics and computer science. In a path-breaking paper based on the non-rigorous ‘cavity method’, physicists predicted not only the existence of a replica symmetry breaking phase transition in such models but also sketched a detailed picture of the evolution of the Gibbs measure within the replica symmetric phase and its impact on important problems in combinatorics, computer science and physics (Krzakala et al. in Proc Natl Acad Sci 104:10318–10323, 2007). In this paper we rigorise this picture completely for a broad class of models, encompassing the Potts antiferromagnet on the random graph, the k-XORSAT model and the diluted k-spin model for even k. We also prove a conjecture about the detection problem in the stochastic block model that has received considerable attention (Decelle et al. in Phys Rev E 84:066106, 2011). PubDate: 2018-02-01 DOI: 10.1007/s00220-018-3096-x

Authors:Claudio Muñoz; Felipe Poblete; Juan C. Pozo Abstract: In this note we show that all small solutions in the energy space of the generalized 1D Boussinesq equation must decay to zero as time tends to infinity, strongly on slightly proper subsets of the space-time light cone. Our result does not require any assumption on the power of the nonlinearity, working even for the supercritical range of scattering. For the proof, we use two new Virial identities in the spirit of works (Kowalczyk et al. in J Am Math Soc 30:769–798, 2017; Kowalczyk et al. in Lett Math Phys 107(5):921–931, 2017). No parity assumption on the initial data is needed. PubDate: 2018-01-31 DOI: 10.1007/s00220-018-3099-7

Authors:Shamit Kachru; Arnav Tripathy Abstract: We show that BPS jumping loci–loci in the moduli space of string compactifications where the number of BPS states jumps in an upper semi-continuous manner—naturally appear as Fourier coefficients of (vector space-valued) automorphic forms. For the case of T2 compactification, the jumping loci are governed by a modular form studied by Hirzebruch and Zagier, while the jumping loci in K3 compactification appear in a story developed by Oda and Kudla–Millson in arithmetic geometry. We also comment on some curious related automorphy in the physics of black hole attractors and flux vacua. PubDate: 2018-01-31 DOI: 10.1007/s00220-018-3090-3

Authors:Benoît Laslier; Fabio Lucio Toninelli Abstract: We study a reversible continuous-time Markov dynamics of a discrete (2 + 1)-dimensional interface. This can be alternatively viewed as a dynamics of lozenge tilings of the \({L\times L}\) torus, or as a conservative dynamics for a two-dimensional system of interlaced particles. The particle interlacement constraints imply that the equilibrium measures are far from being product Bernoulli: particle correlations decay like the inverse distance squared and interface height fluctuations behave on large scales like a massless Gaussian field. We consider a particular choice of the transition rates, originally proposed in Luby et al. (SIAM J Comput 31:167–192, 2001): in terms of interlaced particles, a particle jump of length n that preserves the interlacement constraints has rate 1/(2n). This dynamics presents special features: the average mutual volume between two interface configurations decreases with time (Luby et al. 2001) and a certain one-dimensional projection of the dynamics is described by the heat equation (Wilson in Ann Appl Probab 14:274–325, 2004). In this work we prove a hydrodynamic limit: after a diffusive rescaling of time and space, the height function evolution tends as \({L\to\infty}\) to the solution of a non-linear parabolic PDE. The initial profile is assumed to be C2 differentiable and to contain no “frozen region”. The explicit form of the PDE was recently conjectured (Laslier and Toninelli in Ann Henri Poincaré Theor Math Phys 18:2007–2043, 2017) on the basis of local equilibrium considerations. In contrast with the hydrodynamic equation for the Langevin dynamics of the Ginzburg–Landau model (Funaki and Spohn in Commun Math Phys 85:1–36, 1997; Nishikawa in Commun Math Phys 127:205–227, 2003), here the mobility coefficient turns out to be a non-trivial function of the interface slope. PubDate: 2018-01-30 DOI: 10.1007/s00220-018-3095-y

Authors:Daniel S. Freed; Zohar Komargodski; Nathan Seiberg Abstract: We discuss the three spacetime dimensional \({\mathbb{CP}^N}\) model and specialize to the \({\mathbb{CP}^1}\) model. Because of the Hopf map \({\pi_3(\mathbb{CP}^1)=\mathbb{Z}}\) one might try to couple the model to a periodic θ parameter. However, we argue that only the values θ = 0 and θ = π are consistent. For these values the Skyrmions in the model are bosons and fermions respectively, rather than being anyons. We also extend the model by coupling it to a topological quantum field theory, such that the Skyrmions are anyons. We use techniques from geometry and topology to construct the θ = π theory on arbitrary 3-manifolds, and use recent results about invertible field theories to prove that no other values of \({\theta}\) satisfy the necessary locality. PubDate: 2018-01-30 DOI: 10.1007/s00220-018-3093-0

Authors:Arnaud Brothier; Tobe Deprez; Stefaan Vaes Abstract: We prove the first rigidity and classification theorems for crossed product von Neumann algebras given by actions of non-discrete, locally compact groups. We prove that for arbitrary free probability measure preserving actions of connected simple Lie groups of real rank one, the crossed product has a unique Cartan subalgebra up to unitary conjugacy. We then deduce a W* strong rigidity theorem for irreducible actions of products of such groups. More generally, our results hold for products of locally compact groups that are nonamenable, weakly amenable and that belong to Ozawa’s class \(\mathcal{S}\) . PubDate: 2018-01-30 DOI: 10.1007/s00220-018-3091-2

Authors:D. Beliaev; S. Muirhead Abstract: Smooth random Gaussian functions play an important role in mathematical physics, a main example being the random plane wave model conjectured by Berry to give a universal description of high-energy eigenfunctions of the Laplacian on generic compact manifolds. Our work is motivated by questions about the geometry of such random functions, in particular relating to the structure of their nodal and level sets. We study four discretisation schemes that extract information about level sets of planar Gaussian fields. Each scheme recovers information up to a different level of precision, and each requires a maximum mesh-size in order to be valid with high probability. The first two schemes are generalisations and enhancements of similar schemes that have appeared in the literature (Beffara and Gayet in Publ Math IHES, 2017. https://doi.org/10.1007/s10240-017-0093-0; Mischaikow and Wanner in Ann Appl Probab 17:980–1018, 2007); these give complete topological information about the level sets on either a local or global scale. As an application, we improve the results in Beffara and Gayet (2017) on Russo–Seymour–Welsh estimates for the nodal set of positively-correlated planar Gaussian fields. The third and fourth schemes are, to the best of our knowledge, completely new. The third scheme is specific to the nodal set of the random plane wave, and provides global topological information about the nodal set up to ‘visible ambiguities’. The fourth scheme gives a way to approximate the mean number of excursion domains of planar Gaussian fields. PubDate: 2018-01-29 DOI: 10.1007/s00220-018-3084-1