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:Alexei Davydov; Ana Ros Camacho; Ingo Runkel Abstract: We establish an action of the representations of N = 2-superconformal symmetry on the category of matrix factorisations of the potentials x d and x d −y d , for d odd. More precisely we prove a tensor equivalence between (a) the category of Neveu–Schwarz-type representations of the N = 2 minimal super vertex operator algebra at central charge 3–6/d, and (b) a full subcategory of graded matrix factorisations of the potential x d −y d . The subcategory in (b) is given by permutation-type matrix factorisations with consecutive index sets. The physical motivation for this result is the Landau–Ginzburg/conformal field theory correspondence, where it amounts to the equivalence of a subset of defects on both sides of the correspondence. Our work builds on results by Brunner and Roggenkamp [BR], where an isomorphism of fusion rules was established. PubDate: 2018-02-07 DOI: 10.1007/s00220-018-3086-z

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:Zhiqiang Li Abstract: In this paper, we use the thermodynamical formalism to show that there exists a unique equilibrium state \({\mu_\phi}\) for each expanding Thurston map \({f : S^2\rightarrow S^2}\) together with a real-valued Hölder continuous potential \({\phi}\) . Here the sphere S2 is equipped with a natural metric induced by f, called a visual metric. We also prove that identical equilibrium states correspond to potentials that are co-homologous up to a constant, and that the measure-preserving transformation f of the probability space \({(S^2,\mu_\phi)}\) is exact, and in particular, mixing and ergodic. Moreover, we establish versions of equidistribution of preimages under iterates of f, and a version of equidistribution of a random backward orbit, with respect to the equilibrium state. As a consequence, we recover various results in the literature for a postcritically-finite rational map with no periodic critical points on the Riemann sphere equipped with the chordal metric. PubDate: 2018-01-31 DOI: 10.1007/s00220-017-3073-9

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

Authors:Sourav Chatterjee Abstract: In a celebrated 1990 paper, Aizenman and Wehr proved that the two-dimensional random field Ising model has a unique infinite volume Gibbs state at any temperature. The proof is ergodic-theoretic in nature and does not provide any quantitative information. This article proves the first quantitative version of the Aizenman–Wehr theorem. The proof introduces a new method for proving decay of correlations that may be interesting in its own right. A fairly detailed sketch of the main ideas behind the proof is also included. PubDate: 2018-01-25 DOI: 10.1007/s00220-018-3085-0

Authors:Juan J. L. Velázquez; Raphael Winter Abstract: In this paper, we prove that in macroscopic times of order one, the solutions to the truncated BBGKY hierarchy (to second order) converge in the weak coupling limit to the solution of the nonlinear spatially homogeneous Landau equation. The truncated problem describes the formal leading order behavior of the underlying particle dynamics, and can be reformulated as a non-Markovian hyperbolic equation that converges to the Markovian evolution described by the parabolic Landau equation. The analysis in this paper is motivated by Bogolyubov’s derivation of the kinetic equation by means of a multiple time scale analysis of the BBGKY hierarchy. PubDate: 2018-01-25 DOI: 10.1007/s00220-018-3092-1

Authors:Zhiyuan Zhang Abstract: In this paper, we study the differentiability of SRB measures for partially hyperbolic systems. We show that for any \({s \geq 1}\) , for any integer \({\ell \geq 2}\) , any sufficiently large r, any \({\varphi \in C^{r}(\mathbb{T}, \mathbb{R})}\) such that the map \({f : \mathbb{T}^2 \to \mathbb{T}^2, f(x,y) = (\ell x, y + \varphi(x))}\) is \({C^r}\) -stably ergodic, there exists an open neighbourhood of f in \({C^r(\mathbb{T}^2,\mathbb{T}^2)}\) such that any map in this neighbourhood has a unique SRB measure with \({C^{s-1}}\) density, which depends on the dynamics in a \({C^s}\) fashion. We also construct a \({C^{\infty}}\) mostly contracting partially hyperbolic diffeomorphism \({f: \mathbb{T}^3 \to \mathbb{T}^3}\) such that all f′ in a C2 open neighbourhood of f possess a unique SRB measure \({\mu_{f'}}\) and the map \({f' \mapsto \mu_{f'}}\) is strictly Hölder at f, in particular, non-differentiable. This gives a partial answer to Dolgopyat’s Question 13.3 in Dolgopyat (Commun Math Phys 213:181–201, 2000). PubDate: 2018-01-25 DOI: 10.1007/s00220-018-3088-x

Authors:Maxime Van de Moortel Abstract: We show non-linear stability and instability results in spherical symmetry for the interior of a charged black hole—approaching a sub-extremal Reissner–Nordström background fast enough—in presence of a massive and charged scalar field, motivated by the strong cosmic censorship conjecture in that setting: Stability We prove that spherically symmetric characteristic initial data to the Einstein–Maxwell–Klein–Gordon equations approaching a Reissner–Nordström background with a sufficiently decaying polynomial decay rate on the event horizon gives rise to a space–time possessing a Cauchy horizon in a neighbourhood of time-like infinity. Moreover, if the decay is even stronger, we prove that the space–time metric admits a continuous extension to the Cauchy horizon. This generalizes the celebrated stability result of Dafermos for Einstein–Maxwell-real-scalar-field in spherical symmetry. Instability We prove that for the class of space–times considered in the stability part, whose scalar field in addition obeys a polynomial averaged-L2 (consistent) lower bound on the event horizon, the scalar field obeys an integrated lower bound transversally to the Cauchy horizon. As a consequence we prove that the non-degenerate energy is infinite on any null surface crossing the Cauchy horizon and the curvature of a geodesic vector field blows up at the Cauchy horizon near time-like infinity. This generalizes an instability result due to Luk and Oh for Einstein–Maxwell-real-scalar-field in spherical symmetry. This instability of the black hole interior can also be viewed as a step towards the resolution of the C2 strong cosmic censorship conjecture for one-ended asymptotically flat initial data. PubDate: 2018-01-24 DOI: 10.1007/s00220-017-3079-3

Authors:D. Dragičević; G. Froyland; C. González-Tokman; S. Vaienti Abstract: We prove quenched versions of (i) a large deviations principle (LDP), (ii) a central limit theorem (CLT), and (iii) a local central limit theorem for non-autonomous dynamical systems. A key advance is the extension of the spectral method, commonly used in limit laws for deterministic maps, to the general random setting. We achieve this via multiplicative ergodic theory and the development of a general framework to control the regularity of Lyapunov exponents of twisted transfer operator cocycles with respect to a twist parameter. While some versions of the LDP and CLT have previously been proved with other techniques, the local central limit theorem is, to our knowledge, a completely new result, and one that demonstrates the strength of our method. Applications include non-autonomous (piecewise) expanding maps, defined by random compositions of the form \({T_{\sigma^{n-1} \omega} \circ\cdots\circ T_{\sigma\omega}\circ T_\omega}\) . An important aspect of our results is that we only assume ergodicity and invertibility of the random driving \({\sigma:\Omega\to\Omega}\) ; in particular no expansivity or mixing properties are required. PubDate: 2018-01-19 DOI: 10.1007/s00220-017-3083-7

Authors:Rajesh Kumar Gupta; Sameer Murthy Abstract: We study a class of two-dimensional \({\mathcal{N}=(2,2)}\) sigma models called squashed toric sigma models, using their Gauged Linear Sigma Models (GLSM) description. These models are obtained by gauging the global \({U(1)}\) symmetries of toric GLSMs and introducing a set of corresponding compensator superfields. The geometry of the resulting vacuum manifold is a deformation of the corresponding toric manifold in which the torus fibration maintains a constant size in the interior of the manifold, thus producing a neck-like region. We compute the elliptic genus of these models, using localization, in the case when the unsquashed vacuum manifolds obey the Calabi–Yau condition. The elliptic genera have a non-holomorphic dependence on the modular parameter \({\tau}\) coming from the continuum produced by the neck. In the simplest case corresponding to squashed \({\mathbb{C} / \mathbb{Z}_{2}}\) the elliptic genus is a mixed mock Jacobi form which coincides with the elliptic genus of the \({\mathcal{N}=(2,2)}\) \({SL(2,\mathbb{R}) / U(1)}\) cigar coset. PubDate: 2018-01-18 DOI: 10.1007/s00220-017-3069-5

Authors:Domenico Monaco; Gianluca Panati; Adriano Pisante; Stefan Teufel Abstract: We investigate the localization properties of independent electrons in a periodic background, possibly including a periodic magnetic field, as e. g. in Chern insulators and in quantum Hall systems. Since, generically, the spectrum of the Hamiltonian is absolutely continuous, localization is characterized by the decay, as \({ x \rightarrow \infty}\) , of the composite (magnetic) Wannier functions associated to the Bloch bands below the Fermi energy, which is supposed to be in a spectral gap. We prove the validity of a localization dichotomy in the following sense: either there exist exponentially localized composite Wannier functions, and correspondingly the system is in a trivial topological phase with vanishing Hall conductivity, or the decay of any composite Wannier function is such that the expectation value of the squared position operator, or equivalently of the Marzari–Vanderbilt localization functional, is \({+ \infty}\) . In the latter case, the Bloch bundle is topologically non-trivial, and one expects a non-zero Hall conductivity. PubDate: 2018-01-17 DOI: 10.1007/s00220-017-3067-7

Authors:Matthew Gursky; Casey Lynn Kelleher; Jeffrey Streets Abstract: We show a sharp conformally invariant gap theorem for Yang–Mills connections in dimension 4 by exploiting an associated Yamabe-type problem. PubDate: 2018-01-13 DOI: 10.1007/s00220-017-3070-z