Communications in Mathematical Physics最新文献

We introduce and study a family of power series, which we call Theta series, whose coefficients are in the tensor square of a quantum loop algebra. They arise from a coproduct factorization of the T-series of Frenkel–Hernandez, which are leading terms of transfer matrices of certain infinite-dimensional irreducible modules over the upper Borel subalgebra in the category ({mathcal {O}}) of Hernandez–Jimbo. We prove that each weight component of a Theta series is polynomial. As applications, we establish a decomposition formula and a polynomiality result for R-matrices between an irreducible module and a finite-dimensional irreducible module in category ({mathcal {O}}). We extend T-series and Theta series to Yangians by solving difference equations determined by the truncation series of Gerasimov–Kharchev–Lebedev–Oblezin. We prove polynomiality of Theta series by interpreting them as associators for triple tensor product modules over shifted Yangians.

We show that for any doubling map generated (C^1) monotone potential with derivative uniformly bounded away from zero and infinity, the Lyapunov exponent of the associated Schrödinger operators is bounded below by (log {lambda }-C) for all energies, where C depends only on the potential. In particular, it answers an open question [D, Problem 5] raised by D. Damanik.

In this article, we study wormhole spacetimes in the framework of the static spherically symmetric ({textbf {SU}} (2)) Einstein–Yang–Mills theory coupled to a phantom scalar field. We show rigorously the existence of an infinite sequence of symmetric wormhole solutions, labelled by the number of zeros of the Yang–Mills potential. These solutions have previously been discovered numerically. Mathematically, the problem resembles the pure Einstein–Yang–Mills system for black hole initial conditions, which was well-studied in the 90s. The main difference in the present work is that the coupling to the phantom field adds a non-trivial degree of complexity to the analysis. After proving the existence of the symmetric wormhole solutions, we also present numerical evidence for the existence of asymmetric ones.

A hyperbolic 0-metric on a surface with boundary is a hyperbolic metric on its interior, exhibiting the boundary behavior of the standard metric on the Poincaré disk. Consider the infinite-dimensional Teichmüller spaces of hyperbolic 0-metrics on oriented surfaces with boundary, up to diffeomorphisms fixing the boundary and homotopic to the identity. We show that these spaces have natural symplectic structures, depending only on the choice of an invariant metric on (mathfrak {sl}(2,mathbb {R})). We prove that these Teichmüller spaces are Hamiltonian Virasoro spaces for the action of the universal cover of the group of diffeomorphisms of the boundary. We give an explicit formula for the Hill potential on the boundary defining the moment map. Furthermore, using Fenchel–Nielsen parameters we prove a Wolpert formula for the symplectic form, leading to global Darboux coordinates on the Teichmüller space.

We study surface gravity waves for viscous fluid flows governed by Darcy’s law. The free boundary is acted upon by an external pressure posited to be in traveling wave form with a periodic profile. It has been proven that for any given speed, small external pressures generate small periodic traveling waves that are asymptotically stable. In this work, we construct a class of slowly traveling waves that are of arbitrary size and asymptotically stable. Our results are valid in all dimensions and for both the finite and infinite depth cases.

We prove the almost sure invariance principle (ASIP) with close to optimal error rates for nonuniformly hyperbolic maps. We do not assume exponential contraction along stable leaves, therefore our result covers in particular slowly mixing invertible dynamical systems as Bunimovich flowers, billiards with flat points as in Chernov and Zhang (Stoch Dyn 5:535–553, 2005a, Nonlinearity 18:1527–1553, 2005b) and Wojtkowski’ (Commun Math Phys 126:507–533, 1990) system of two falling balls.For these examples, the ASIP is a new result, not covered by prior works for various reasons, notably because in absence of exponential contraction along stable leaves, it is challenging to employ the so-called Sinai’s trick (Sinai in Russ Math Surv 27:21–70, 1972; Bowen, Lecture Notes in Math vol. 470 (1975)) of reducing a nonuniformly hyperbolic system to a nonuniformly expanding one. Our strategy follows our previous papers on the ASIP for nonuniformly expanding maps, where we build a semiconjugacy to a specific renewal Markov shift and adapt the argument of Berkes et al. (Ann Probab 42:794–817, 2014). The main difference is that now the Markov shift is two-sided, the observables depend on the full trajectory, both the future and the past.

We study the nonlinear dynamics of perturbed, spectrally stable T-periodic stationary solutions of the Lugiato–Lefever equation (LLE), a damped nonlinear Schrödinger equation with forcing that arises in nonlinear optics. It is known that for each (Nin {mathbb {N}}), such a T-periodic wave train is asymptotically stable against NT-periodic, i.e. subharmonic, perturbations, in the sense that initially nearby data will converge at an exponential rate to a (small) spatial translation of the underlying wave. Unfortunately, in such results both the allowable size of initial perturbations as well as the exponential rates of decay depend on N and, in fact, tend to zero as (Nrightarrow infty ), leading to a lack of uniformity in the period of the perturbation. In recent work, the authors performed a delicate decomposition of the associated linearized solution operator and obtained linear estimates which are uniform in N. The dynamical description suggested by this uniform linear theory indicates that the corresponding nonlinear iteration can only be closed if one allows for a spatio-temporal phase modulation of the underlying wave. However, such a modulated perturbation is readily seen to satisfy a quasilinear equation, yielding an inherent loss of regularity. We regain regularity by transferring a nonlinear damping estimate, which has recently been obtained for the LLE in the case of localized perturbations to the case of subharmonic perturbations. Thus, we obtain a nonlinear, subharmonic stability result for periodic stationary solutions of the LLE that is uniform in N. This in turn yields an improved nonuniform subharmonic stability result providing an N-independent ball of initial perturbations which eventually exhibit exponential decay at an N-dependent rate. Finally, we argue that our results connect in the limit (N rightarrow infty ) to previously established stability results against localized perturbations, thereby unifying existing theories.

The Potts spin glass is a generalization of the Sherrington–Kirkpatrick (SK) model that allows for spins to take more than two values. Based on a novel synchronization mechanism, Panchenko (Ann Probab 46(2):829–864, 2018) showed that the limiting free energy is given by a Parisi-type variational formula. The functional order parameter in this formula is a probability measure on a monotone path in the space of positive-semidefinite matrices. By comparison, the order parameter for the SK model is much simpler: a probability measure on the unit interval. Nevertheless, a longstanding prediction by Elderfield and Sherrington (J Phys C Solid State Phys 16(15):L497–L503, 1983) is that the order parameter for the Potts spin glass can be reduced to that of the SK model. We prove this prediction for the balanced Potts spin glass, where the model is constrained so that the fraction of spins taking each value is asymptotically the same. It is generally believed that the limiting free energy of the balanced model is the same as that of the unconstrained model, in which case our results reduce the functional order parameter of Panchenko’s variational formula to probability measures on the unit interval. The intuitive reason—for both this belief and the Elderfield–Sherrington prediction—is that no spin value is a priori preferred over another, and the order parameter should reflect this inherent symmetry. This paper rigorously demonstrates how symmetry, when combined with synchronization, acts as the desired reduction mechanism. Our proof requires that we introduce a generalized Potts spin glass model with mixed higher-order interactions, which is interesting it its own right. We prove that the Parisi formula for this model is differentiable with respect to inverse temperatures. This is a key ingredient for guaranteeing the Ghirlanda–Guerra identities without perturbation, which then allow us to exploit symmetry and synchronization simultaneously.

We study the free energy of a mean-field spin glass whose coupling distribution has power-law tails. For couplings with infinite variance and finite mean, we show that the thermodynamic limit of the quenched free energy exists and that the free energy is self-averaging.

We prove the existence and uniqueness of solution of the loop equation associated with a semisimple generalized Frobenius manifold with non-flat unity, and show, for a particular example of one-dimensional generalized Frobenius manifold, that the deformation of the Principal Hierarchy induced by the solution of the loop equation is the extended q-deformed KdV hierarchy.