Random wave functions and percolation

Recently it was conjectured that nodal domains of random wave functions are adequately described by critical percolation theory. In this paper we strengthen this conjecture in two respects. First, we show that, though wave function correlations decay slowly, a careful use of Harris' criterion confirms that these correlations are unessential and nodal domains of random wave functions belong to the same universality class as non critical percolation. Second, we argue that level domains of random wave functions are described by the non-critical percolation model.Comment: 13 page

Critical holes in undercooled wetting layers

The profile of a critical hole in an undercooled wetting layer is determined by the saddle-point equation of a standard interface Hamiltonian supported by convenient boundary conditions. It is shown that this saddle-point equation can be mapped onto an autonomous dynamical system in a three-dimensional phase space. The corresponding flux has a polynomial form and in general displays four fixed points, each with different stability properties. On the basis of this picture we derive the thermodynamic behaviour of critical holes in three different nucleation regimes of the phase diagram.Comment: 18 pages, LaTeX, 6 figures Postscript, submitted to J. Phys.

The Statistics of the Points Where Nodal Lines Intersect a Reference Curve

We study the intersection points of a fixed planar curve $\Gamma$ with the nodal set of a translationally invariant and isotropic Gaussian random field \Psi(\bi{r}) and the zeros of its normal derivative across the curve. The intersection points form a discrete random process which is the object of this study. The field probability distribution function is completely specified by the correlation G(|\bi{r}-\bi{r}'|) = . Given an arbitrary G(|\bi{r}-\bi{r}'|), we compute the two point correlation function of the point process on the line, and derive other statistical measures (repulsion, rigidity) which characterize the short and long range correlations of the intersection points. We use these statistical measures to quantitatively characterize the complex patterns displayed by various kinds of nodal networks. We apply these statistics in particular to nodal patterns of random waves and of eigenfunctions of chaotic billiards. Of special interest is the observation that for monochromatic random waves, the number variance of the intersections with long straight segments grows like $L \ln L$, as opposed to the linear growth predicted by the percolation model, which was successfully used to predict other long range nodal properties of that field.Comment: 33 pages, 13 figures, 1 tabl

Chaos and stability in a two-parameter family of convex billiard tables

We study, by numerical simulations and semi-rigorous arguments, a two-parameter family of convex, two-dimensional billiard tables, generalizing the one-parameter class of oval billiards of Benettin--Strelcyn [Phys. Rev. A 17, 773 (1978)]. We observe interesting dynamical phenomena when the billiard tables are continuously deformed from the integrable circular billiard to different versions of completely-chaotic stadia. In particular, we conjecture that a new class of ergodic billiard tables is obtained in certain regions of the two-dimensional parameter space, when the billiards are close to skewed stadia. We provide heuristic arguments supporting this conjecture, and give numerical confirmation using the powerful method of Lyapunov-weighted dynamics.Comment: 19 pages, 13 figures. Submitted for publication. Supplementary video available at http://sistemas.fciencias.unam.mx/~dsanders

On the Nodal Count Statistics for Separable Systems in any Dimension

We consider the statistics of the number of nodal domains aka nodal counts for eigenfunctions of separable wave equations in arbitrary dimension. We give an explicit expression for the limiting distribution of normalised nodal counts and analyse some of its universal properties. Our results are illustrated by detailed discussion of simple examples and numerical nodal count distributions.Comment: 21 pages, 4 figure

Nodal domains on quantum graphs

We consider the real eigenfunctions of the Schr\"odinger operator on graphs, and count their nodal domains. The number of nodal domains fluctuates within an interval whose size equals the number of bonds $B$. For well connected graphs, with incommensurate bond lengths, the distribution of the number of nodal domains in the interval mentioned above approaches a Gaussian distribution in the limit when the number of vertices is large. The approach to this limit is not simple, and we discuss it in detail. At the same time we define a random wave model for graphs, and compare the predictions of this model with analytic and numerical computations.Comment: 19 pages, uses IOP journal style file

Universal Fluctuations in Correlated Systems

The probability density function (PDF) of a global measure in a large class of highly correlated systems has been suggested to be of the same functional form. Here, we identify the analytical form of the PDF of one such measure, the order parameter in the low temperature phase of the 2D-XY model. We demonstrate that this function describes the fluctuations of global quantities in other correlated, equilibrium and non-equilibrium systems. These include a coupled rotor model, Ising and percolation models, models of forest fires, sand-piles, avalanches and granular media in a self organized critical state. We discuss the relationship with both Gaussian and extremal statistics.Comment: 4 pages, 2 figure

Magnetization distribution in the transverse Ising chain with energy flux

The zero-temperature transverse Ising chain carrying an energy flux j_E is studied with the aim of determining the nonequilibrium distribution functions, P(M_z) and P(M_x), of its transverse and longitudinal magnetizations, respectively. An exact calculation reveals that P(M_z) is a Gaussian both at j_E=0 and j_E not equal 0, and the width of the distribution decreases with increasing energy flux. The distribution of the order-parameter fluctuations, P(M_x), is evaluated numerically for spin-chains of up to 20 spins. For the equilibrium case (j_E=0), we find the expected Gaussian fluctuations away from the critical point while the critical order-parameter fluctuations are shown to be non-gaussian with a scaling function Phi(x)=Phi(M_x/)=P(M_x) strongly dependent on the boundary conditions. When j_E not equal 0, the system displays long-range, oscillating correlations but P(M_x) is a Gaussian nevertheless, and the width of the Gaussian decreases with increasing j_E. In particular, we find that, at critical transverse field, the width has a j_E^(-3/8) asymptotic in the j_E -> 0 limit.Comment: 8 pages, 5 ps figure

Statistics of extremal intensities for Gaussian interfaces

The extremal Fourier intensities are studied for stationary Edwards-Wilkinson-type, Gaussian, interfaces with power-law dispersion. We calculate the probability distribution of the maximal intensity and find that, generically, it does not coincide with the distribution of the integrated power spectrum (i.e. roughness of the surface), nor does it obey any of the known extreme statistics limit distributions. The Fisher-Tippett-Gumbel limit distribution is, however, recovered in three cases: (i) in the non-dispersive (white noise) limit, (ii) for high dimensions, and (iii) when only short-wavelength modes are kept. In the last two cases the limit distribution emerges in novel scenarios.Comment: 15 pages, including 7 ps figure

Width Distributions and the Upper Critical Dimension of KPZ Interfaces

Simulations of restricted solid-on-solid growth models are used to build the width-distributions of d=2-5 dimensional KPZ interfaces. We find that the universal scaling function associated with the steady-state width-distribution changes smoothly as d is increased, thus strongly suggesting that d=4 is not an upper critical dimension for the KPZ equation. The dimensional trends observed in the scaling functions indicate that the upper critical dimension is at infinity.Comment: 4 pages, 4 postscript figures, RevTe