We study nodal sets for typical eigenfunctions of the Laplacian on the standard torus in 2 or more dimensions. Making use of the multiplicities in the spectrum of the Laplacian, we put a Gaussian measure on the eigenspaces and use it to average over the eigenspace. We consider a sequence of eigenvalues with multiplicity N tending to infinity. The quantity that we study is the Leray, or microcanonical, measure of the nodal set. We show that the expected value of the Leray measure of an eigenfunction is constant. Our main result is that the variance of Leray measure is asymptotically 1 (4 pi N), as N tends to infinity, at least in dimensions 2 and at least 5.
The study of nodal lines of random waves goes back to Longuet-Higgins @cite_7 @cite_13 who computed various statistics of nodal lines for Gaussian random waves in connection with the analysis of ocean waves. Berry @cite_10 suggested to model highly excited quantum states for classically chaotic systems by using various random wave models, and also computed fluctuations of various quantities in these models (see e.g. @cite_2 ). See also Zelditch @cite_11 . The idea of averaging over a single eigenspace in the presence of multiplicities appears in B 'erard @cite_17 who computed the expected surface measure of the nodal set for eigenfunctions of the Laplacian on spheres. Neuheisel @cite_9 also worked on the sphere and studied the statistics of Leray measure. He gave an upper bound for the variance, which we believe is not sharp.
@cite_13 @cite_7 @cite_9 @cite_17 @cite_2 @cite_10 @cite_11
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For real (time-reversal symmetric) quantum billiards, the mean length L of nodal line is calculated for the nth mode (n>>1), with wavenumber k, using a Gaussian random wave model adapted locally to satisfy Dirichlet or Neumann boundary conditions. The leading term is of order k (i.e. √n), and the first (perimeter) correction, dominated by an unanticipated long-range boundary effect, is of order log k (i.e. log n), with the same sign (negative) for both boundary conditions. The leading-order state-to-state fluctuations δL are of order √log k. For the curvature κ of nodal lines, |κ| and √κ2 are of order k, but |κ|3 and higher moments diverge. For complex (e.g. Aharonov-Bohm) billiards, the mean number N of nodal points (phase singularities) in the mode has a leading term of order k2 (i.e. n), the perimeter correction (again a long-range effect) is of order klog k (i.e. √nlog n) (and positive, notwithstanding nodal depletion near the boundary) and the fluctuations δN are of order k√log k. Generalizations of the results for mixed (Robin) boundary conditions are stated. Complex arithmetic random waves are stationary Gaussian complex-valued solutions of the Helmholtz equation on the two-dimensional flat torus. We use Wiener-It ^o chaotic expansions in order to derive a complete characterization of the second order high-energy behaviour of the total number of phase singularities of these functions. Our main result is that, while such random quantities verify a universal law of large numbers, they also exhibit non-universal and non-central second order fluctuations that are dictated by the arithmetic nature of the underlying spectral measures. Such fluctuations are qualitatively consistent with the cancellation phenomena predicted by Berry (2002) in the case of complex random waves on compact planar domains. Our results extend to the complex setting recent pathbreaking findings by Rudnick and Wigman (2008), Krishnapur, Kurlberg and Wigman (2013) and Marinucci, Peccati, Rossi and Wigman (2016). The exact asymptotic characterization of the variance is based on a fine analysis of the Kac-Rice kernel around the origin, as well as on a novel use of combinatorial moment formulae for controlling long-range weak correlations. We study individual eigenstates of quantized area-preserving maps on the 2-torus which are classically chaotic. In order to analyze their semiclassical behavior, we use the Bargmann–Husimi representations for quantum states as well as their stellar parametrization, which encodes states through a minimal set of points in phase space (the constellation of zeros of the Husimi density). We rigorously prove that a semiclassical uniform distribution of Husimi densities on the torus entails a similar equidistribution for the corresponding constellations. We deduce from this property a universal behavior for the phase patterns of chaotic Bargmann eigenfunctions which is reminiscent of the WKB approximation for eigenstates of integrable systems (though in a weaker sense). In order to obtain more precise information on “chaotic eigenconstellations,” we then model their properties by ensembles of random states, generalizing former results on the 2-sphere to the torus geometry. This approach yields statistical predictions for the constellations which fit quite well the chaotic data. We finally observe that specific dynamical information, e.g., the presence of high peaks (like scars) in Husimi densities, can be recovered from the knowledge of a few long-wavelength Fourier coefficients, which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions. We study the asymptotic statistical behavior of the 2-dimensional periodic Lorentz gas with an infinite horizon. We consider a particle moving freely in the plane with elastic reflections from a periodic set of fixed convex scatterers. We assume that the initial position of the particle in the phase space is random with uniform distribution with respect to the Liouville measure of the periodic problem. We are interested in the asymptotic statistical behavior of the particle displacement in the plane as the timet goes to infinity. We assume that the particle horizon is infinite, which means that the length of free motion of the particle is unbounded. Then we show that under some natural assumptions on the free motion vector autocorrelation function, the limit distribution of the particle displacement in the plane is Gaussian, but the normalization factor is (t logt)1 2 and nott1 2 as in the classical case. We find the covariance matrix of the limit distribution.