Topological constraints in geometric deformation quantization on domains with multiple boundary components

A topological constraint on the possible values of the universal quantization parameter is revealed in the case of geometric quantization on (boundary) curves diffeomorphic to $S^1$, analytically extended on a bounded domain in $\mathbb{C}$, with $n \ge 2$ boundary components. Unlike the case of one boundary component (such as the canonical Berezin quantization of the Poincar\'e upper-half plane or the case of conformally-invariant 2D systems), the more general case considered here leads to a strictly positive minimum value for the quantization parameter, which depends on the geometrical data of the domain (specifically, the total area and total perimeter in the smooth case). It is proven that if the lower bound is attained, then $n=2$ and the domain must be annular, with a direct interpretation in terms of the global monodromy

Universal limits of nonlinear measure redistribution processes and their applications

Deriving the time evolution of a distribution of probability (or a probability density matrix) is a problem encountered frequently in a variety of situations: for physical time, it could be a kinetic reaction study, while identifying time with the number of computational steps gives a typical picture of algorithms routinely used in quantum impurity solvers, density functional theory, etc. Using a truncation scheme for the expansion of the exact quantity is necessary due to constraints of the numerical implementation. However, this leads in turn to serious complications such as the Fermion Sign Problem (essentially, density or weights will become negative). By integrating angular degrees of freedom and reducing the dynamics to the radial component, the time evolution is reformulated as a nonlinear integral transform of the distribution function. A canonical decomposition into orthogonal polynomials leads back to the original sign problem, but using a characteristic-function representation allows to extract the asymptotic behavior, and gives an exact large-time limit, for many initial conditions, with guaranteed positivity.Comment: To appear in the Journal of Problems of Nonlinear Analysis in Engineering System

Efficient algorithms for topological inference on random graphs

In this study, we investigate the problem of classifying, characterizing, and designing efficient algorithms for hard inference problems on planar graphs, in the limit of infinite size. The problem is considered hard if, for a deterministic graph, it belongs to the NP class of computational complexity. A typical example rich in applications is that of connectivity loss in evacuation models for natural hazards management (e.g. coastal floods, hurricanes). Algorithmically, this model reduces to solving a min-cut (or max-flow) problem, with is known to be intractable. The current work covers several generalizations: posing the same problem for non-directed networks subject to random fluctuations (specifically, random graphs from the Erd\"os-R\'enyi class); finding efficient convex classifiers for the associated decision problem (deciding whether the graph had become disconnected or not); and the role played by choice of topology (on the space of random graphs) in designing efficient, convex approximation algorithms (in the infinite-size limit of the graph)

Effective distribution of codewords for Low Density Parity Check Cycle codes in the presence of disorder

We review the zeta-function representation of codewords allowed by a parity-check code based on a bipartite graph, and then investigate the effect of disorder on the effective distribution of codewords. The randomness (or disorder) is implemented by sampling the graph from an ensemble of random graphs, and computing the average zeta function of the ensemble. In the limit of arbitrarily large size for the vertex set of the graph, we find an exponential decay of the likelihood for nontrivial codewords corresponding to graph cycles. This result provides a quantitative estimate of the effect of randomization in cybersecurity applications

Braid group representations and cold Fermi gases in the fast pairing regime

It is widely recognized that the main difficulty in designing devices which could process information using quantum states is due to the decoherence of local excitations about a ground state. A solution to this problem was suggested in \cite{Kitaev}, relying on (non-local) topological excitations, structurally protected against local noise. However, a practical implementation of this proposal using special Landau levels in fractional quantum Hall effect systems (FQHE) \cite{QHE} has proven elusive, while accessible FQHE states are theoretically not optimal because their representations in the Hilbert space of states are not dense. We propose using a different physical system (cold Fermi atoms), whose semiclassical dynamics is described by a hyperelliptic function in the Sklyanin formalism. The homological structure of the complex curve corresponds to representations of the braid group, with the action of Hecke operators leading to singularities detectable in the semiclassical oscillations. We argue that, for a fixed genus of the hyperelliptic curve, the Richardson-Gaudin pairing Hamiltonian problem is the singular limit $m \to \infty$ of level-$k$ $\widehat{sl}(2)$, with $k+2 = \frac{4}{8m+1}\to 0$, so that the level $k$ is admissible in the sense of Kac and Kazhdan \cite{KK}, but the corresponding Hecke algebra is a $q-$deformation of the symmetric group with fixed $q = e^{i\pi/4}$, as $m \to \infty$.Comment: arXiv admin note: text overlap with arXiv:0712.356

Coherent oscillations in superconducting cold Fermi atoms and their applications

Recent achievements in experiments with cold fermionic atoms indicate the potential for developing novel superconducting devices which may be operated in a wide range of regimes, at a level of precision previously not available. Unlike traditional, solid-state superconducting devices, the cold-atom systems allow the fast switching on of the BCS phase, and the observation of non-equilibrium, coherent oscillations of the order parameter. The integrable and non-linear nature of the equations of motions makes this operating regime particularly rich in potential applications, such as quantum modulation and encoding, or nonlinear mixing of quantum coherent oscillations, to name only two. From a mathematical point of view, such systems can be described using the Knizhnik-Zamolodchikov-Bernard equation, or more generally, by the matrix Kadomtsev-Petviashvilii integrable hierarchy. This identification is particularly useful, since it allows a direct application of the known non-linear phenomena described by particular solutions of these equations. Other important features of this formulation, such as the relation to the spin Calogero-Sutherland model, also have relevant physical interpretations. In this work, a complete description of these relationships is presented, along with their potential practical consequences.Comment: Invited chapter in "Leading-edge Superconductivity Research Developments", Nov

Weak solution of the Hele-Shaw problem: shocks and viscous fingering

In Hele-Shaw flows, boundaries between fluids develop unstable viscous fingers. At vanishing surface tension, the fingers further evolve to cusp-like singularities. We show that the problem admits a {\it weak solution} where shock fronts triggered by a singularity propagate together with a fluid. Shocks form a growing, branching tree of a mass deficit, and a line distribution of vorticity where pressure and velocity of the fluid have finite discontinuities. Imposing that the flow remain curl-free at macroscale determines the shock graph structure. We present a self-similar solution describing shocks emerging from a generic (2,3)-cusp singularity -- an elementary branching event.Comment: 5 pages, 3 figures. To appear in the Journal of Experimental and Theoretical Physics, JETP Letter

Optimal approximation of harmonic growth clusters by orthogonal polynomials

Interface dynamics in two-dimensional systems with a maximal number of conservation laws gives an accurate theoretical model for many physical processes, from the hydrodynamics of immiscible, viscous flows (zero surface-tension limit of Hele-Shaw flows, [1]), to the granular dynamics of hard spheres [2], and even diffusion-limited aggregation [3]. Although a complete solution for the continuum case exists [4, 5], efficient approximations of the boundary evolution are very useful due to their practical applications [6]. In this article, the approximation scheme based on orthogonal polynomials with a deformed Gaussian kernel [7] is discussed, as well as relations to potential theory

Non-equilibrium thermodynamics for functionals of current and density

We study a stochastic many-body system maintained in an non-equilibrium steady state. Probability distribution functional of the time-integrated current and density is shown to attain a large-deviation form in the long-time asymptotics. The corresponding Current-Density Cramer Functional (CDCF) is explicitly derived for irreversible Langevin dynamics and discrete-space Markov chains. We also show that the Cramer functionals of other linear functionals of density and current, like work generated by a force, are related to CDCF in a way reminiscent of variational relations between different thermodynamic potentials. The general formalism is illustrated with a model example.Comment: Submitted to Phys. Rev. Let

Around a theorem of F. Dyson and A. Lenard: Energy Equilibria for Point Charge Distributions in Classical Electrostatics

We discuss several results in electrostatics: Onsager's inequality, an extension of Earnshaw's theorem, and a result stemming from the celebrated conjecture of Maxwell on the number of points of electrostatic equilibrium. Whenever possible, we try to provide a brief historical context and references.Comment: 14 page