We investigate the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a Hermitian Wigner matrix at the edge of the spectrum. We show that the suitably rescaled second-order correlation function is asymptotically given by the Airy kernel, thereby generalizing the well-known result for the Gaussian Unitary Ensemble (GUE). Moreover, we obtain similar ...

We analyse new data for self-avoiding polygons (SAPs), on the square and triangular lattices, enumerated by both perimeter and area, providing evidence that the scaling function is the logarithm of an Airy function. The results imply universal amplitude combinations for all area moments and suggest that rooted SAPs may satisfy a q-algebraic functional equation. PACS numbers: 05.50+q, 02.10.AB, ...

We report a high power ultrafast Airy beam source producing femtosecond pulses at multi-gigahertz (GHz) repetition rate (RR). Based on intra-cavity cubic phase modulation of an optical parametric oscillator (OPO) designed in high harmonic cavity configuration synchronous to a femtosecond Yb-fiber laser operating at 78 MHz, we have produced ultrafast 2D Airy beam at multi-GHz repetition rate thr...

The Airy process τ → Aτ is characterized by its finite-dimensional distribution functions Pr (Aτ1 < ξ1, . . . , Aτm < ξm) . For m = 1 it is known that Pr (Aτ < ξ) is expressible in terms of a solution to Painlevé II. We show that each finite-dimensional distribution function is expressible in terms of a solution to a system of differential equations.

Computer algebra algorithms are developed for evaluating the coefficients in Airy-type asymptotic expansions that are obtained from integrals with a large parameter. The coefficients are defined from recursive schemes obtained from integration by parts. An application is given for the Weber parabolic cylinder function. 1991 Mathematics Subject Classification: 41A60, 33C10, 33C15, 33F05, 65D20.

ABSTRACT Modified Airy Function (MAF) method gives an approximate solution of the wave equation for planar waveguides with an arbitrary refractive index distribution. Unlike WKB method, the MAF solution does not diverge at the turning points and there is no need of connection formulae. Comparison with exact results for practical wave guides show that the errors in MAF solutions are small. Use o...

For a plane wave incident on either a Luneburg lens or a modified Luneburg lens, the magnitude and phase of the transmitted electric field are calculated as a function of the scattering angle in the context of ray theory. It is found that the ray trajectory and the scattered intensity are not uniformly convergent in the vicinity of edge ray incidence on a Luneburg lens, which corresponds to the...

Integral representations are onsidered of solutions of the Airy di erential equation w z w = 0 for omputing Airy fun tions for omplex values of z. In a rst method ontour integral representations of the Airy fun tions are written as non-os illating integrals for obtaining stable representations, whi h are evaluated by the trapezoidal rule. In a se ond method an integral representation is evaluat...

We establish the universal edge scaling limit of random partitions with infinite-parameter distribution called Schur measure. explore asymptotic behavior wave function, which is a building block corresponding kernel, based on Schrödinger-type differential equation. show that function in general to Airy and its higher-order analogs limit. construct kernel Tracy–Widom from discuss implication mul...

We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process A(y). The Airy process is stationary, it has continuous sample paths, its single “time” (fixed y) distribution is the Tracy-Widom distribution of the largest eigenvalue of a GUE random matrix, and the Airy process has ...