FORMULAE AND ASYMPTOTICS FOR COEFFICIENTS OF ALGEBRAIC FUNCTIONS

International audienc

Asymptotics of Bivariate Generating Functions with Algebraic Singularities

Flajolet and Odlyzko (1990) derived asymptotic formulae the coefficients of a class of uni- variate generating functions with algebraic singularities. Gao and Richmond (1992) and Hwang (1996, 1998) extended these results to classes of multivariate generating functions, in both cases by reducing to the univariate case. Pemantle and Wilson (2013) outlined new multivariate ana- lytic techniques and used them to analyze the coefficients of rational generating functions. After overviewing these methods, we use them to find asymptotic formulae for the coefficients of a broad class of bivariate generating functions with algebraic singularities. Beginning with the Cauchy integral formula, we explicity deform the contour of integration so that it hugs a set of critical points. The asymptotic contribution to the integral comes from analyzing the integrand near these points, leading to explicit asymptotic formulae. Next, we use this formula to analyze an example from current research. In the following chapter, we apply multivariate analytic techniques to quan- tum walks. Bressler and Pemantle (2007) found a (d + 1)-dimensional rational generating function whose coefficients described the amplitude of a particle at a position in the integer lattice after n steps. Here, the minimal critical points form a curve on the (d + 1)-dimensional unit torus. We find asymptotic formulae for the amplitude of a particle in a given position, normalized by the number of steps n, as n approaches infinity. Each critical point contributes to the asymptotics for a specific normalized position. Using Groebner bases in Maple again, we compute the explicit locations of peak amplitudes. In a scaling window of size the square root of n near the peaks, each amplitude is asymptotic to an Airy function

Geometry of q-Hypergeometric Functions as a Bridge between Yangians and Quantum Affine Algebras

The rational quantized Knizhnik-Zamolodchikov equation (qKZ equation) associated with the Lie algebra $sl_2$ is a system of linear difference equations with values in a tensor product of $sl_2$ Verma modules. We solve the equation in terms of multidimensional $q$-hypergeometric functions and define a natural isomorphism between the space of solutions and the tensor product of the corresponding quantum group $U_q(sl_2)$ Verma modules, where the parameter $q$ is related to the step $p$ of the qKZ equation via $q=e^{pi i/p}$. We construct asymptotic solutions associated with suitable asymptotic zones and compute the transition functions between the asymptotic solutions in terms of the trigonometric $R$-matrices. This description of the transition functions gives a new connection between representation theories of Yangians and quantum loop algebras and is analogous to the Kohno-Drinfeld theorem on the monodromy group of the differential Knizhnik-Zamolodchikov equation. In order to establish these results we construct a discrete Gauss-Manin connection, in particular, a suitable discrete local system, discrete homology and cohomology groups with coefficients in this local system, and identify an associated difference equation with the qKZ equation.Comment: 66 pages, amstex.tex (ver. 2.1) and amssym.tex are required; misprints are correcte

Non-Weyl Resonance Asymptotics for Quantum Graphs

We consider the resonances of a quantum graph $\mathcal G$ that consists of a compact part with one or more infinite leads attached to it. We discuss the leading term of the asymptotics of the number of resonances of $\mathcal G$ in a disc of a large radius. We call $\mathcal G$ a \emph{Weyl graph} if the coefficient in front of this leading term coincides with the volume of the compact part of $\mathcal G$. We give an explicit topological criterion for a graph to be Weyl. In the final section we analyze a particular example in some detail to explain how the transition from the Weyl to the non-Weyl case occurs.Comment: 29 pages, 2 figure

Eigenvalue problem in a solid with many inclusions: asymptotic analysis

We construct the asymptotic approximation to the first eigenvalue and corresponding eigensolution of Laplace's operator inside a domain containing a cloud of small rigid inclusions. The separation of the small inclusions is characterised by a small parameter which is much larger compared with the nominal size of inclusions. Remainder estimates for the approximations to the first eigenvalue and associated eigenfield are presented. Numerical illustrations are given to demonstrate the efficiency of the asymptotic approach compared to conventional numerical techniques, such as the finite element method, for three-dimensional solids containing clusters of small inclusions.Comment: 55 pages, 5 figure

Short-distance thermal correlations in the massive XXZ chain

We explore short-distance static correlation functions in the infinite XXZ chain using previously derived formulae which represent the correlation functions in factorized form. We compute two-point functions ranging over 2, 3 and 4 lattice sites as functions of the temperature and the magnetic field in the massive regime $\Delta>1$, extending our previous results to the full parameter plane of the antiferromagnetic chain ($\Delta > -1$ and arbitrary field $h$). The factorized formulae are numerically efficient and allow for taking the isotropic limit ($\Delta = 1$) and the Ising limit ($\Delta = \infty$). At the critical field separating the fully polarized phase from the N\'eel phase, the Ising chain possesses exponentially many ground states. The residual entropy is lifted by quantum fluctuations for large but finite $\Delta$ inducing unexpected crossover phenomena in the correlations.Comment: 24 pages, color onlin