The multi-instanton expansion for the eigenvalues of the symmetric double well is derived using a Langer–Cherry uniform asymptotic expansion of the solution of the corresponding Schrödinger equation. The Langer–Cherry expansion is anchored to either one of the minima of the potential, and by construction has the correct asymptotic behavior at large distance, while the quantization condition amounts to imposing the even or odd parity of the wave function. This method leads to an efficient algorithm for the calculation to virtually any desired order of all the exponentially small series of the multi-instanton expansion, and with trivial modifications can also be used for nonsymmetric double wells.

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We show analytically that the perturbative expansion for the free energy of the zero dimensional (quenched) disordered Ising model is Borel-summable in a certain range of parameters, provided that the summation is carried out in two steps: first, in the strength of the original coupling of the Ising model and subsequently in the variance of the quenched disorder. This result is illustrated by some high-precision calculations of the free energy obtained by a straightforward numerical implementation of our sequential summation method.

This paper deals with the determination of the S-curves in the theory of non-hermitian orthogonal polynomials with respect to exponential weights along suitable paths in the complex plane. It is known that the corresponding complex equilibrium potential can be written as a combination of Abelian integrals on a suitable Riemann surface whose branch points can be taken as the main parameters of the problem. Equations for these branch points can be written in terms of periods of Abelian differentials and are known in several equivalent forms. We select one of these forms and use a combination of analytic an numerical methods to investigate the phase structure of asymptotic zero densities of orthogonal polynomials and of asymptotic eigenvalue densities of random matrix models. As an application we give a complete description of the phases and critical processes of the standard cubic model.

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Abstract. We present an implementation of the method of orthogonal polynomials which is particularly suitable to study the partition functions of Penner random matrix models, to obtain their explicit forms in the exactly solvable cases, and to determine the coefficients of their perturbative expansions in the continuum limit. The method relies on identities satisfied by the resolvent of the Jacobi matrix in the three-term recursion relation of the associated families of orthogonal polynomials. These identities lead to a convenient formulation of the string equations. As an application, we show that in the continuum limit the free energy of certain exactly solvable models like the linear and double Penner models can be written as a sum of gaussian contributions plus linear terms. To illustrate the one-cut case we discuss the linear, double and cubic Penner models, and for the two-cut case we discuss theoretically and numerically the existence of a double-branch structure of the free energy for the gaussian Penner model.

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We experimentally demonstrate a sigmoidal variation of the composition profile across semiconductor heterointerfaces. The wide range of material systems (III-arsenides, III-antimonides, III-V quaternary compounds, III-nitrides) exhibiting such a profile suggests a universal behavior. We show that sigmoidal profiles emerge from a simple model of cooperative growth mediated by twodimensional island formation, wherein cooperative effects are described by a specific functional dependence of the sticking coefficient on the surface coverage. Experimental results confirm that, except in the very early stages, island growth prevails over nucleation as the mechanism governing the interface development and ultimately determines the sigmoidal shape of the chemical profile in these two-dimensional grown layers. In agreement with our experimental findings, the model also predicts a minimum value of the interfacial width, with the minimum attainable value depending on the chemical identity of the species. A central goal of modern materials physics is the control of interfaces down to the atomic level. In particular, the behavior of layered materials depends on the atomicscale structural roughness and chemical mixing across the interface [1]. Although abrupt interfaces between conventional semiconductors (such as III-V compounds) are fabricated and element profiles across these interfaces are obtained with atomic resolution, the relation between the layer growth processes and the parameters governing the interface formation and evolution is not satisfactorily understood. In this respect, there is an ongoing discussion about how interfaces can be quantitatively described on the basis of a growth model and whether there is a minimum interface width.Recently Hulko et al. [2,3] and have shown empirically that experimental concentration profiles in III-V two-dimensional (2D) heterostructures, e.g. quantum wells (QW) grown by molecular beam epitaxy (MBE), can be accurately reproduced by a sigmoidal function of the formHere, x 0 denotes the nominal mole fraction of one of the species, z is the position across the interface along the growth direction, and L is the parameter quantifying the interface width (L is proportional to the widely reported length W , over which the concentration changes from 10% to 90% of its plateau value). Moreover, the accuracy of the sigmoidal fitting seems to be independent of the experimental technique used to obtain the element distribution [2,5,7] and, more interestingly, of the compound semiconductor. In this letter, we show that a sigmoidal profile emerges from a simple model of cooperative growth with 2D island formation. Furthermore, the use of a generalized sigmoidal expression gives a reliable and systematic quantification of the chemical interface. It sheds light on basic aspects of the early stages of heteroepitaxial growth, and permits to find a correlation between the profile and the interface properties in morphologically perfect epitaxial layers [8], which have been grown in the thermodynam...