We calculate analytically, for finite-size matrices, joint probability densities of ratios of level spacings in ensembles of random matrices characterized by their associated confining potential. We focus on the ratios of two spacings between three consecutive real eigenvalues, as well as certain generalizations such as the overlapping ratios. The resulting formulas are further analyzed in detail in two specific cases: the β-Hermite and the β-Laguerre cases, for which we offer explicit calculations for small N . The analytical results are in excellent agreement with numerical simulations of usual random matrix ensembles, and with the level statistics of a quantum many-body lattice model and zeros of the Riemann zeta function.

We point out that the transmission eigenvalue density and higher order correlation functions in chaotic cavities for an arbitrary number of incoming and outgoing leads (N 1 , N 2 ) are analytically known from the Jacobi ensemble of Random Matrix Theory. Using this result and a simple linear statistic, we give an exact and non-perturbative expression for moments of the form λ m 1 for m > −|N 1 − N 2 | − 1 and β = 2, thus improving the existing results in the literature. Secondly, we offer an independent derivation of the average density and higher order correlation functions for β = 2, 4 which does not make use of the orthogonal polynomials technique. This result may be relevant for an efficient numerical implementation avoiding determinants.

Among systems that display generic scale invariance, those whose asymptotic properties are anisotropic in space (strong anisotropy, SA) have received relatively less attention, especially in the context of kinetic roughening for two-dimensional surfaces. This is in contrast with their experimental ubiquity, e.g., in the context of thin-film production by diverse techniques. Based on exact results for integrable (linear) cases, here we formulate a SA ansatz that, albeit equivalent to existing ones borrowed from equilibrium critical phenomena, is more naturally adapted to the type of observables that are measured in experiments on the dynamics of thin films, such as one- and two-dimensional height structure factors. We test our ansatz on a paradigmatic nonlinear stochastic equation displaying strong anisotropy like the Hwa-Kardar equation [Phys. Rev. Lett. 62, 1813 (1989)], which was initially proposed to describe the interface dynamics of running sand piles. A very important role to elucidate its SA properties is played by an accurate (Gaussian) approximation through a nonlocal linear equation that shares the same asymptotic properties.

We report an experimental assessment of surface kinetic roughening properties that are anisotropic in space. Working for two specific instances of silicon surfaces irradiated by ion-beam sputtering under diverse conditions (with and without concurrent metallic impurity codeposition), we verify the predictions and consistency of a recently proposed scaling Ansatz for surface observables like the two-dimensional (2D) height power spectral density (PSD). In contrast with other formulations, this ansatz is naturally tailored to the study of two-dimensional surfaces, and allows us to readily explore the implications of anisotropic scaling for other observables, such as real-space correlation functions and PSD functions for 1D profiles of the surface. Our results confirm that there are indeed actual experimental systems whose kinetic roughening is strongly anisotropic, as consistently described by this scaling analysis. In the light of our work, some types of experimental measurements are seen to be more affected by issues like finite space resolution effects, etc. that may hinder a clear-cut assessment of strongly anisotropic scaling in the present and other practical contexts.

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