Scaled Correlations of Critical Points of Random Sections on Riemann Surfaces

In this paper we prove that as N goes to infinity, the scaling limit of the correlation between critical points z1 and z2 of random holomorphic sections of the N-th power of a positive line bundle over a compact Riemann surface tends to 2/(3pi^2) for small sqrt(N)|z1-z2|. The scaling limit is directly calculated using a general form of the Kac-Rice formula and formulas and theorems of Pavel Bleher, Bernard Shiffman, and Steve Zelditch.Comment: 55 pages. LaTeX. output.txt is the output of running heisenberg_simpler.mpl through maple. heisenberg_simpler.mpl can be run by maple at the command line by saying 'maple -q heisenberg_simpler.mpl' to see the maple calculations that generated the matrices U(t) and D(t) described in the paper's appendix. It may also be run by opening it with GUI mapl

Semi-classical analysis of non self-adjoint transfer matrices in statistical mechanics. I

We propose a way to study one-dimensional statistical mechanics models with complex-valued action using transfer operators. The argument consists of two steps. First, the contour of integration is deformed so that the associated transfer operator is a perturbation of a normal one. Then the transfer operator is studied using methods of semi-classical analysis. In this paper we concentrate on the second step, the main technical result being a semi-classical estimate for powers of an integral operator which is approximately normal.Comment: 28 pp, improved the presentatio

Asymptotic Laws for the Spatial Distribution and the Number of Connected Components of Zero Sets of Gaussian Random Functions

We study the asymptotic laws for the spatial distribution and the number of connected components of zero sets of smooth Gaussian random functions of several real variables. The primary examples are various Gaussian ensembles of real-valued polynomials (algebraic or trigonometric) of large degree on the sphere or torus, and translation-invariant smooth Gaussian functions on the Euclidean space restricted to large domains

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

The density of states of 1D random band matrices via a supersymmetric transfer operator

Recently, T. and M. Shcherbina proved a pointwise semicircle law for the density of states of one-dimensional Gaussian band matrices of large bandwidth. The main step of their proof is a new method to study the spectral properties of non-self-adjoint operators in the semiclassical regime. The method is applied to a transfer operator constructed from the supersymmetric integral representation for the density of states. We present a simpler proof of a slightly upgraded version of the semicircle law, which requires only standard semiclassical arguments and some peculiar elementary computations. The simplification is due to the use of supersymmetry, which manifests itself in the commutation between the transfer operator and a family of transformations of superspace, and was applied earlier in the context of band matrices by Constantinescu. Other versions of this supersymmetry have been a crucial ingredient in the study of the localization–delocalization transition by theoretical physicists

Zeroes of Gaussian Analytic Functions with Translation-Invariant Distribution

We study zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We prove that the a limiting horizontal mean counting-measure of the zeroes exists almost surely, and that it is non-random if and only if the spectral measure is continuous (or degenerate). In this case, the mean zero-counting measure is computed in terms of the spectral measure. We compare the behavior with Gaussian analytic function with symmetry around the real axis. These results extend a work by Norbert Wiener.Comment: 24 pages, 1 figure. Some corrections were made and presentation was improve