Communications in Mathematical Physics最新文献

We formulate a family of algebras, twisted Yangians (of split type) in current generators and relations, via a degeneration of the Drinfeld presentation of affine (imath )quantum groups (associated with split Satake diagrams). These new algebras admit PBW type bases and are shown to be a deformation of twisted current algebras; presentations for twisted current algebras are also provided. For type AI, it matches with the Drinfeld presentation of twisted Yangian obtained via Gauss decomposition. We conjecture that our split twisted Yangians are isomorphic to the corresponding ones in RTT presentation.

When two spatially separated parties make measurements on an unknown entangled quantum state, what correlations can they achieve? How difficult is it to determine whether a given correlation is a quantum correlation? These questions are central to problems in quantum communication and computation. Previous work has shown that the general membership problem for quantum correlations is computationally undecidable. In the current work we show something stronger: there is a family of constant-sized correlations—that is, correlations for which the number of measurements and number of measurement outcomes are fixed—such that solving the quantum membership problem for this family is computationally impossible. Thus, the undecidability that arises in understanding Bell experiments is not dependent on varying the number of measurements in the experiment. This places strong constraints on the types of descriptions that can be given for quantum correlation sets. Our proof is based on a combination of techniques from quantum self-testing and undecidability results for linear system nonlocal games.

We use the theory of (x-y) duality to propose a new definition/construction for the correlation differentials of topological recursion; we call it generalized topological recursion. This new definition coincides with the original topological recursion of Chekhov–Eynard–Orantin in the regular case and allows, in particular, to get meaningful answers in a variety of irregular and degenerate situations.

We study quantum neural networks made by parametric one-qubit gates and fixed two-qubit gates in the limit of infinite width, where the generated function is the expectation value of the sum of single-qubit observables over all the qubits. First, we prove that the probability distribution of the function generated by the untrained network with randomly initialized parameters converges in distribution to a Gaussian process whenever each measured qubit is correlated only with few other measured qubits. Then, we analytically characterize the training of the network via gradient descent with square loss on supervised learning problems. We prove that, as long as the network is not affected by barren plateaus, the trained network can perfectly fit the training set and that the probability distribution of the function generated after training still converges in distribution to a Gaussian process. Finally, we consider the statistical noise of the measurement at the output of the network and prove that a polynomial number of measurements is sufficient for all the previous results to hold and that the network can always be trained in polynomial time.

A rigorous derivation of point vortex systems from kinetic equations has been a challenging open problem, due to singular layers in the inviscid limit, giving a large velocity gradient in the Boltzmann equations. In this paper, we derive the Helmholtz–Kirchhoff point-vortex system from the hydrodynamic limits of the Boltzmann equations. We construct Boltzmann solutions by the Hilbert-type expansion associated to the point vortices solutions of the 2D Navier–Stokes equations. We give a precise pointwise estimate for the solution of the Boltzmann equations with small Strouhal number and Knudsen number.

We study the geometry of integrable systems of hydrodynamic type of the form (w_t=Xcirc w_x) where (circ ) is the product of a regular F-manifold. In the first part of the paper, we present a general construction of a connection compatible with the F-manifold structure starting from a frame of vector fields defining commuting flows of hydrodynamic type. In the second part of the paper, using this construction, we study regular F-manifolds with compatible connection and Euler vector field, ((nabla ,circ ,e,E)), associated with integrable hierarchies obtained from the solutions of the equation (dcdot d_L ,a_0=0) where (L=Ecirc ). In particular, we show that n-dimensional F-manifolds associated to regular operators L are classified by n arbitrary functions of a single variable. Moreover, we show that flat connections (nabla ) correspond to linear solutions (a_0).

Diffusion limited aggregation and its generalization, dielectric-breakdown model play an important role in physics, approximating a range of natural phenomena. Yet little is known about them, with the famous Kesten’s estimate on the DLAs growth being perhaps the most important result. Using a different approach we prove a generalisation of this result for the DBM in (mathbb {Z}^2) and (mathbb {Z}^3). The obtained estimate depends on the DBM parameter, and matches with the best known results for DLA. In particular, since our methods are different from Kesten’s, our argument provides a new proof for Kesten’s result both in (mathbb {Z}^2) and (mathbb {Z}^3).

This work is devoted to systematically study general N-soliton solutions possibly containing multiple degenerate soliton groups (DSGs), in the context of the sharp-line Maxwell–Bloch equations with a zero background. We also show that results can be readily migrated to other integrable systems, with the same non-self-adjoint Zakharov–Shabat scattering problem or alike. Results for the focusing nonlinear Schrödinger equation and the complex modified Korteweg–De Vries equation are obtained as explicit examples for demonstrative purposes. A DSG is a localized coherent nonlinear traveling-wave structure, comprised of inseparable solitons with identical velocities. Hence, DSGs are generalizations of single solitons (considered as 1-DSGs), and form fundamental building blocks of solutions of many integrable systems. We provide an explicit formula for an N-DSG and its center. With the help of the Deift–Zhou’s nonlinear steepest descent method, we prove the localization of DSGs, and calculate the long-time asymptotics for an arbitrary N-soliton solutions. It is shown that the solution becomes a linear combination of multiple DSGs in the distant past and future, with explicit formulæ for the asymptotic phase shift for each DSG. Other generalizations of a single soliton are also discussed, such as Nth-order solitons and soliton gases. We prove that every Nth-order soliton can be obtained by fusion of eigenvalues of N-soliton solutions, with proper rescalings of norming constants, and demonstrate that soliton-gas solution can be considered as limits of N-soliton solutions as (Nrightarrow +infty ).

We study the Glauber dynamics for heavy-tailed spin glasses, in which the couplings are in the domain of attraction of an (alpha )-stable law for (alpha in (0,1)). We show a sharp description of metastability on exponential timescales, in a form that is believed to hold for Glauber/Langevin dynamics for many mean-field spin glass models, but only known rigorously for the Random Energy Models. Namely, we establish a decomposition of the state space into sub-exponentially many wells, and show that the projection of the Glauber dynamics onto which well it resides in, asymptotically behaves like a Markov chain on wells with certain explicit transition rates. In particular, mixing inside wells occurs on much shorter timescales than transit times between wells, and the law of the next well the Glauber dynamics will fall into depends only on which well it currently resides in, not its full configuration. We can deduce consequences like an exact expression for the two-time autocorrelation functions that appear in the activated aging literature.