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Vesicles, or closed fluctuating membranes, have been modelled in two dimensions by self-avoiding polygons, weighted with respect to their perimeter and enclosed area, with the simplest model given by area-weighted excursions (Dyck paths). These models generically show a tricritical phase transition between an inflated and a crumpled phase, with a scaling function given by the logarithmic derivative of the Airy function. Extending such a model, we find realisations of multicritical points of arbitrary order, with the associated multivariate scaling functions expressible in terms of generalised Airy integrals, as previously conjectured by John Cardy. This work therefore adds to the small list of models with a critical phase transition, for which exponents and the associated scaling functions are explicitly known. [ABSTRACT FROM AUTHOR]

We study the higher-order heat-type equation with first time and Mth spatial partial derivatives, M = 2, 3, ... . We demonstrate that its exact solutions for M even can be constructed with the help of signed Lévy stable functions. For M odd the same role is played by a special generalization of the Airy Ai function that we introduce and study. This permits one to generate the exact and explicit heat kernels pertaining to these equations. We examine analytically and graphically the spatial and temporary evolution of particular solutions for simple initial conditions. [ABSTRACT FROM AUTHOR]

In a recent contribution, Dotsenko establishes a Fredholm determinant formula for the two-point distribution of the Kardar-Parisi-Zhang equation in the long time limit and starting from narrow wedge initial conditions. We establish that his expression is identical to the Fredholm determinant resulting from the Airy2 process. [ABSTRACT FROM AUTHOR]

We produce a family of matrix Riemann-Hilbert (RH) problems parametrized by a given scalar function r(k). The jump matrix of the RH problem depends on r(k) and external real parameters x and t. We prove the unique solvability of such a problem. A solution M(x, t, k) of this matrix RH problem gives a smooth solution q(x, t ) of the modified Korteweg-de Vries (mKdV) equation qt + 6q2qx + qxxx = 0. For a special choice of the function r(k) we show that the corresponding initial data q(x, 0) is a Heaviside-type function. We consider the small-time limit of the solution q(x, t ). The leading-order term is explicitly calculated as an integral of the Airy function … The first summand is an exact solution to the linear KdV equation pt +pxxx = 0 with initial data … Thus this specific solution q(x, t ) to the mKdV equation, being smooth with respect to x &egr; R and t > 0, takes pointwise its limit values q(x, 0) = limt→+0 q(x, t ) = p(x) and exhibits a Gibbs-type phenomenon with rapid oscillations near x = 0 as t→+0. [ABSTRACT FROM AUTHOR]

Kontsevich's work on Airy matrix integrals has led to explicit results for the intersection numbers of the moduli space of curves. In a subsequent work Okounkov rederived these results from the edge behavior of a Gaussian matrix integral. In our work we consider the correlation functions of vertices in a Gaussian random matrix theory, with an external matrix source. We deal with operator products of the form \big\langle \prod_{i=1}^n \frac{1}{N}\,{\rm tr}\, M^{k_i}\big\rangle , in a \frac{1}{N} expansion. For large values of the powers ki, in an appropriate scaling limit relating large k's to large N, universal scaling functions are derived. Furthermore, we show that the replica method applied to characteristic polynomials of the random matrices, together with a duality exchanging N and the number of points, provides a new way to recover Kontsevich's results on these intersection numbers. [ABSTRACT FROM AUTHOR]

The q-difference Painlevé II equation admits special solutions written in terms of determinants whose entries are the general solution of the q-Airy equation. An ultradiscrete limit of the special solutions is studied using the procedure of ultradiscretization with parity variables. Then we obtain new Airy-type solutions of the ultradiscrete Painlevé II equation with parity variables, and the solutions have a richer structure than the known solutions. [ABSTRACT FROM AUTHOR]

Several recent works have considered the pressure exerted on a wall by a model polymer. We extend this consideration to vesicles attached to a wall, and hence include osmotic pressure. We do this by considering a two-dimensional directed model, namely that of area-weighted Dyck paths. Not surprisingly, the pressure exerted by the vesicle on the wall depends on the osmotic pressure inside, especially its sign. Here, we discuss the scaling of this pressure in the different regimes, paying particular attention to the crossover between positive and negative osmotic pressure. In our directed model, there exists an underlying Airy function scaling form, from which we extract the dependence of the bulk pressure on small osmotic pressures. [ABSTRACT FROM AUTHOR]

We consider the asymptotic expansion of the logarithmic derivative of the Airy function Ai'(z)/Ai(z), and also its reciprocal Ai(z)/Ai'(z), as |z| - [?]. We derive simple, closed-form solutions for the coefficients which appear in these expansions, which are of interest since they are encountered in a wide variety of problems. The solutions are presented as Mellin transforms of given functions; this fact, together with the methods employed, suggests further avenues for research. [ABSTRACT FROM AUTHOR]

We derive new constraints on the zeros of Airy functions by using the so-called quantum bouncer system to evaluate quantum-mechanical sum rules and perform perturbation theory calculations for the Stark effect. Using commutation and completeness relations, we show how to systematically evaluate sums of the form Sp(n) = [?]k[?]n1/(zk [?] zn)p, for natural p > 1, where [?]zn is the nth zero of Ai(z). [ABSTRACT FROM AUTHOR]