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Abstract: We introduce a shell model of turbulence featuring intermittent behaviour with anomalous power-law scaling of structure functions. This model is solved analytically with the explicit derivation of anomalous exponents. The solution associates the intermittency with the hidden symmetry for Kolmogorov multipliers, making our approach relevant for real turbulence. PubDate: 2021-11-01
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Abstract: We analyse the eigenvectors of the adjacency matrix of a critical Erdős–Rényi graph \({\mathbb {G}}(N,d/N)\) , where d is of order \(\log N\) . We show that its spectrum splits into two phases: a delocalized phase in the middle of the spectrum, where the eigenvectors are completely delocalized, and a semilocalized phase near the edges of the spectrum, where the eigenvectors are essentially localized on a small number of vertices. In the semilocalized phase the mass of an eigenvector is concentrated in a small number of disjoint balls centred around resonant vertices, in each of which it is a radial exponentially decaying function. The transition between the phases is sharp and is manifested in a discontinuity in the localization exponent \(\gamma (\varvec{\mathrm {w}})\) of an eigenvector \(\varvec{\mathrm {w}}\) , defined through \(\Vert \varvec{\mathrm {w}} \Vert _\infty / \Vert \varvec{\mathrm {w}} \Vert _2 = N^{-\gamma (\varvec{\mathrm {w}})}\) . Our results remain valid throughout the optimal regime \(\sqrt{\log N} \ll d \leqslant O(\log N)\) . PubDate: 2021-11-01
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Abstract: The monster sporadic group is the automorphism group of a central charge \(c=24\) vertex operator algebra (VOA) or meromorphic conformal field theory (CFT). In addition to its \(c=24\) stress tensor T(z), this theory contains many other conformal vectors of smaller central charge; for example, it admits 48 commuting \(c=\frac{1}{2}\) conformal vectors whose sum is T(z). Such decompositions of the stress tensor allow one to construct new CFTs from the monster CFT in a manner analogous to the Goddard-Kent-Olive (GKO) coset method for affine Lie algebras. We use this procedure to produce evidence for the existence of a number of CFTs with sporadic symmetry groups and employ a variety of techniques, including Hecke operators, modular linear differential equations, and Rademacher sums, to compute the characters of these CFTs. Our examples include (extensions of) nine of the sporadic groups appearing as subquotients of the monster, as well as the simple groups \({}^2{\textit{E}}_6(2)\) and \({\textit{F}}_4(2)\) of Lie type. Many of these examples are naturally associated to McKay’s \(\widehat{E_8}\) correspondence, and we use the structure of Norton’s monstralizer pairs more generally to organize our presentation. PubDate: 2021-11-01
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Abstract: Let \(0<\sigma <n/2\) and \(H=(-\Delta )^\sigma +V(x)\) be Schrödinger type operators on \({\mathbb {R}}^n\) with a class of scaling-critical potentials V(x), which include the Hardy potential \(a x ^{-2\sigma }\) with a sharp coupling constant \(a\ge -C_{\sigma ,n}\) ( \(C_{\sigma ,n}\) is the best constant of Hardy’s inequality of order \(\sigma \) ). In the present paper we consider several sharp global estimates for the resolvent and the solution to the time-dependent Schrödinger equation associated with H. In the case of the subcritical coupling constant \(a>-C_{\sigma ,n}\) , we first prove uniform resolvent estimates of Kato–Yajima type for all \(0<\sigma <n/2\) , which turn out to be equivalent to Kato smoothing estimates for the Cauchy problem. We then establish Strichartz estimates for \(\sigma >1/2\) and uniform Sobolev estimates of Kenig–Ruiz–Sogge type for \(\sigma \ge n/(n+1)\) . These extend the same properties for the Schrödinger operator with the inverse-square potential to the higher-order and fractional cases. Moreover, we also obtain improved Strichartz estimates with a gain of regularities for general initial data if \(1<\sigma <n/2\) and for radially symmetric data if \(n/(2n-1)<\sigma \le 1\) , which extends the corresponding results for the free evolution to the case with Hardy potentials. These arguments can be further applied to a large class of higher-order inhomogeneous elliptic operators and even to certain long-range metric perturbations of the Laplace operator. Finally, in the critical coupling constant case (i.e., \(a=-C_{\sigma ,n}\) ), we show that the same results as in the subcritical case still hold for functions orthogonal to radial functions. PubDate: 2021-11-01
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Abstract: In this paper, we show the limiting absorption principle for the wave operator on asymptotically Minkowski spacetimes. This problem was previously considered by Vasy (J Spectr Theory 10:439–461, 2020). Here, we employ Mourre theory which seems a more transparent tool. Moreover, we also prove that the anti-Feynman propagator defined by Gérard and Wrochna coincides with the outgoing resolvent. PubDate: 2021-11-01
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Abstract: We apply the technique of convex integration to obtain non-uniqueness and existence results for power-law fluids, in dimension \(d\ge 3\) . For the power index q below the compactness threshold, i.e. \(q \in (1, \frac{2d}{d+2})\) , we show ill-posedness of Leray–Hopf solutions. For a wider class of indices \(q \in (1, \frac{3d+2}{d+2})\) we show ill-posedness of distributional (non-Leray–Hopf) solutions, extending the seminal paper of Buckmaster & Vicol [10]. In this wider class we also construct non-unique solutions for every datum in \(L^2\) . PubDate: 2021-11-01
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Abstract: We define the double ramification hierarchy associated to an F-cohomological field theory and use this construction to prove that the principal hierarchy of any semisimple (homogeneous) flat F-manifold possesses a (homogeneous) integrable dispersive deformation at all orders in the dispersion parameter. The proof is based on the reconstruction of an F-CohFT starting from a semisimple flat F-manifold and additional data in genus 1, obtained in our previous work. Our construction of these dispersive deformations is quite explicit and we compute several examples. In particular, we provide a complete classification of rank 1 hierarchies of DR type at the order 9 approximation in the dispersion parameter and of homogeneous DR hierarchies associated with all 2-dimensional homogeneous flat F-manifolds at genus 1 approximation. PubDate: 2021-11-01
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Abstract: We provide a new large class of countable icc groups \({\mathcal {A}}\) for which the product rigidity result from Chifan et al. (Geom Funct Anal 26(1): 136–159, 2016) holds: if \(\Gamma _1,\ldots ,\Gamma _n\in {\mathcal {A}}\) and \(\Lambda \) is any group such that \(L(\Gamma _1\times \dots \times \Gamma _n)\cong L(\Lambda )\) , then there exists a product decomposition \(\Lambda =\Lambda _1\times \dots \times \Lambda _n\) such that \(L(\Lambda _i)\) is stably isomorphic to \(L(\Gamma _i)\) , for any \(1\le i\le n\) . Class \({\mathcal {A}}\) consists of groups \(\Gamma \) for which \(L(\Gamma )\) admits an s-malleable deformation in the sense of Sorin Popa and it includes all non-amenable groups \(\Gamma \) such that either (a) \(\Gamma \) admits an unbounded 1-cocycle into its left regular representation, or (b) \(\Gamma \) is an arbitrary wreath product group with amenable base. As a byproduct of these results, we obtain new examples of W \(^*\) -superrigid groups and new rigidity results in the C \(^*\) -algebra theory. PubDate: 2021-11-01
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Abstract: For a group G and \(\omega \in Z^{3}(G, \text {U}(1))\) , an \(\omega \) -anomalous action on a C*-algebra B is a \(\text {U}(1)\) -linear monoidal functor between 2-groups , where the latter denotes the 2-group of \(*\) -automorphisms of B. The class \([\omega ]\in H^{3}(G, \text {U}(1))\) is called the anomaly of the action. We show that for every \(n\ge 2\) and every finite group G, every anomaly can be realized on the stabilization of a commutative C*-algebra \(C(M)\otimes {\mathcal {K}}\) for some closed connected n-manifold M. We also show that although there are no anomalous symmetries of Roe C*-algebras of coarse spaces, for every finite group G, every anomaly can be realized on the Roe corona \(C^{*}(X)/{\mathcal {K}}\) of some bounded geometry metric space X with property A. PubDate: 2021-11-01
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Abstract: We present a new, general approach to gauge theory on principal G-spectral triples, where G is a compact connected Lie group. We introduce a notion of vertical Riemannian geometry for G- \(C^*\) -algebras and prove that the resulting noncommutative orbitwise family of Kostant’s cubic Dirac operators defines a natural unbounded \(KK^G\) -cycle in the case of a principal G-action. Then, we introduce a notion of principal G-spectral triple and prove, in particular, that any such spectral triple admits a canonical factorisation in unbounded \(KK^G\) -theory with respect to such a cycle: up to a remainder, the total geometry is the twisting of the basic geometry by a noncommutative superconnection encoding the vertical geometry and underlying principal connection. Using these notions, we formulate an approach to gauge theory that explicitly generalises the classical case up to a groupoid cocycle and is compatible in general with this factorisation; in the unital case, it correctly yields a real affine space of noncommutative principal connections with affine gauge action. Our definitions cover all locally compact classical principal G-bundles and are compatible with \(\theta \) -deformation; in particular, they cover the \(\theta \) -deformed quaternionic Hopf fibration \(C^\infty (S^7_\theta ) \hookleftarrow C^\infty (S^4_\theta )\) as a noncommutative principal \({{\,\mathrm{SU}\,}}(2)\) -bundle. PubDate: 2021-11-01
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Abstract: Persistent topological structures in physical systems have become increasingly important over the last years. Electromagnetic fields with knotted field lines play a special role among these, since they can be used to transfer their knottedness to other systems like plasmas and quantum fluids. In null electromagnetic fields the electric and the magnetic field lines evolve like unbreakable elastic filaments in a fluid flow. In particular, their topology is preserved for all time, so that all knotted closed field lines maintain their knot type. We use an approach due to Bateman to prove that for every link L there is such an electromagnetic field that satisfies Maxwell’s equations in free space and that has closed electric and magnetic field lines in the shape of L for all time. The knotted and linked field lines turn out to be projections of real analytic Legendrian links with respect to the standard contact structure on the 3-sphere. PubDate: 2021-11-01
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Abstract: In the strong noise regime, we study the homogenization of quantum trajectories i.e. stochastic processes appearing in the context of quantum measurement. When the generator of the average semigroup can be separated into three distinct time scales, we start by describing a homogenized limiting semigroup. This result is of independent interest and is formulated outside of the scope of quantum trajectories. Going back to the quantum context, we show that, in the Meyer–Zheng topology, the time-continuous quantum trajectories converge weakly to the discontinuous trajectories of a pure jump Markov process. Notably, this convergence cannot hold in the usual Skorokhod topology. PubDate: 2021-11-01
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Abstract: We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity \(\lambda \beta e^{\beta u }\) , forced by an additive space-time white noise. (i) We first study SNLH for general \(\lambda \in {\mathbb {R}}\) . By establishing higher moment bounds of the relevant Gaussian multiplicative chaos and exploiting the positivity of the Gaussian multiplicative chaos, we prove local well-posedness of SNLH for the range \(0< \beta ^2 < \frac{8 \pi }{3 + 2 \sqrt{2}} \simeq 1.37 \pi \) . Our argument yields stability under the noise perturbation, thus improving Garban’s local well-posedness result (2020). (ii) In the defocusing case \(\lambda >0\) , we exploit a certain sign-definite structure in the equation and the positivity of the Gaussian multiplicative chaos. This allows us to prove global well-posedness of SNLH for the range: \(0< \beta ^2 < 4\pi \) . (iii) As for SdNLW in the defocusing case \(\lambda > 0\) , we go beyond the Da Prato-Debussche argument and introduce a decomposition of the nonlinear component, allowing us to recover a sign-definite structure for a rough part of the unknown, while the other part enjoys a stronger smoothing property. As a result, we reduce SdNLW into a system of equations (as in the paracontrolled approach for the dynamical \(\Phi ^4_3\) -model) and prove local well-posedness of SdNLW for the range: \(0< \beta ^2 < \frac{32 - 16\sqrt{3}}{5}\pi \simeq 0.86\pi \) . This result (translated to the context of random data well-posedness for the deterministic nonlinear wave equation with an exponential nonlinearity) solves an open question posed by Sun and Tzvetkov (2020). (iv) When \(\lambda > 0\) , these models formally preserve the associated Gibbs measures with the exponential nonlinearity. Under the same assumption on \(\beta \) as in (ii) and (iii) above, we prove almost sure global well-posedness (in particular for SdNLW) and invariance of the Gibbs measures in both the parabolic and hyperbolic settings. (v) In Appendix, we present an argument for proving local well-posedness of SNLH for general \(\lambda \in {\mathbb {R}}\) without using the positivity of the Gaussian multiplicative chaos. This proves local well-posedness of SNLH for the range \(0< \beta ^2 < \frac{4}{3} \pi \simeq 1.33 \pi \) , slightly smaller than that in (i), but provides Lipschitz continuity of the solution map in initial data as well as the noise. PubDate: 2021-11-01
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Abstract: In this paper, we investigate the geometry of the base complex manifold of an effectively parametrized holomorphic family of stable Higgs bundles over a fixed compact Kähler manifold. The starting point of our study is Schumacher–Toma/Biswas–Schumacher’s curvature formulas for Weil–Petersson-type metrics, in Sect. 2, we give some applications of their formulas on the geometric properties of the base manifold. In Sect. 3, we calculate the curvature on the higher direct image bundle, which recovers Biswas–Schumacher’s curvature formula. In Sect. 4, we construct a smooth and strongly pseudo-convex complex Finsler metric for the base manifold, the corresponding holomorphic sectional curvature is calculated explicitly. PubDate: 2021-11-01
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Abstract: The stability of solutions under periodic perturbations for both inviscid and viscous conservation laws is an interesting and important problem. In this paper, a large-amplitude viscous shock under space-periodic perturbation for the isentropic Navier–Stokes equations is considered. It is shown that if the initial perturbation around the shock is suitably small and satisfies a zero-mass type condition (2.17), then the solution of the N–S equations tends to the viscous shock with a shift, which is partially determined by the periodic oscillations. In other words, the viscous shock is nonlinearly stable even though the perturbation oscillates at the far fields. The key point is to construct a suitable ansatz \( (\tilde{v},\tilde{u}) \) , which carries the same oscillations of the solution (v, u) at the far fields, so that the difference \( (v-\tilde{v},u-\tilde{u}) \) belongs to the \( H^2(\mathbb {R}) \) space for all \( t\ge 0. \) PubDate: 2021-11-01
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Abstract: Let M be a Riemannian, boundaryless, and compact manifold, with \(\dim M\ge 2\) and let f be a \(C^{1+}\) diffeomorphism. We show that there is a hyperbolic SRB measure if and only if there exists an unstable leaf with a subset of positive leaf volume of hyperbolic points which return to some Pesin set with positive frequency. This answers a question of Pesin. PubDate: 2021-11-01
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Abstract: We derive a new Hamiltonian formulation of Schlesinger equations in terms of the dynamical r-matrix structure. The corresponding symplectic form is shown to be the pullback, under the monodromy map, of a natural symplectic form on the extended monodromy manifold. We show that Fock–Goncharov coordinates are log-canonical for the symplectic form. Using these coordinates we define the symplectic potential on the monodromy manifold and interpret the Jimbo–Miwa–Ueno tau-function as the generating function of the monodromy map. This, in particular, solves a recent conjecture by A. Its, O. Lisovyy and A. Prokhorov. PubDate: 2021-10-12
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Abstract: We construct an infinite volume spatial random permutation \(({{\textsf {X}}},\sigma )\) , where \({{\textsf {X}}}\subset {\mathbb {R}}^d\) is locally finite and \(\sigma :{{\textsf {X}}}\rightarrow {{\textsf {X}}}\) is a permutation, associated to the formal Hamiltonian $$\begin{aligned} H({{\textsf {X}}},\sigma ) = \sum _{x\in {{\textsf {X}}}} \Vert x-\sigma (x)\Vert ^2. \end{aligned}$$ The measures are parametrized by the point density \(\rho \) and the temperature \(\alpha \) . Spatial random permutations are naturally related to boson systems through a representation originally due to Feynman (Phys Rev 91:1291–1301, 1953). Let \(\rho _c=\rho _c(\alpha )\) be the critical density for Bose–Einstein condensation in Feynman’s representation. Each finite cycle of \(\sigma \) induces a loop of points of \({{\textsf {X}}}\) . For \(\rho \le \rho _c\) we define \(({{\textsf {X}}}, \sigma )\) as a Poisson process of finite unrooted loops of a random walk with Gaussian increments that we call Gaussian loop soup, analogous to the Brownian loop soup of Lawler and Werner (Probab Theory Related Fields 128(4):565–588, 2004). We also construct Gaussian random interlacements, a Poisson process of doubly infinite trajectories of random walks with Gaussian increments analogous to the Brownian random interlacements of Sznitman (Ann Math 2 171(3):2039–2087, 2010). For \(d\ge 3\) and \(\rho >\rho _c\) we define \(({{\textsf {X}}},\sigma )\) as the superposition of independent realizations of the Gaussian loop soup at density \(\rho _c\) and the Gaussian random interlacements at density \(\rho -\rho _c\) . In either case we call \(({{\textsf {X}}}, \sigma )\) a Gaussian random permutation at density \(\rho \) and temperature \(\alpha \) . The resulting measure satisfies a Markov property and it is Gibbs for the Hamiltonian H. Its point marginal \({{\textsf {X}}}\) has the same distribution as the boson point process introduced by Shirai-Takahashi (J Funct Anal 205(2):414–463, 2003) in the subcritical case, and by Tamura-Ito (J Funct Anal 243(1): 207–231, 2007) in the supercritical case. PubDate: 2021-10-04
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