Gaussian Fluctuation in Random Matrices

Abstract

Let $N(L)$ be the number of eigenvalues, in an interval of length $L$, of a matrix chosen at random from the Gaussian Orthogonal, Unitary or Symplectic ensembles of ${\cal N}$ by ${\cal N}$ matrices, in the limit ${\cal N}\rightarrow\infty$. We prove that $[N(L) - \langle N(L)\rangle]/\sqrt{\log L}$ has a Gaussian distribution when $L\rightarrow\infty$. This theorem, which requires control of all the higher moments of the distribution, elucidates numerical and exact results on chaotic quantum systems and on the statistics of zeros of the Riemann zeta function. \noindent PACS nos. 05.45.+b, 03.65.-wComment: 13 page

Anisotropic Growth of Random Surfaces in 2 + 1 Dimensions

“…More precisely, in Section 2 of [50] the convergence in distribution (a generalization of the result for the sine kernel of [18]) is stated. However, following the proof of the theorem, one realizes that it is done by controlling the cumulants, i.e., also the moments converge.…”

“…Finally, let us mention that our proof of Theorem 1.1 is based on the argument of [18] and [50], the proof of Theorem 1.3 uses several ideas from [33], and the algebraic formalism for two-dimensional growth models employs a crucial idea of constructing bivariate Markov chains out of commuting univariate ones from [19].…”

Lower entropy bounds and particle number fluctuations in a Fermi sea

“…At zero temperature the Fermi sea defines a projection P E , such as projection on momentum states below k F , or the lowest Landau level in the example below. One may then replace M by the operator P E P A P E in all expressions such as (10), (12) and (13). Note that whenever the average number of particles in the box A is finite, the operator P E P A P E has a finite trace, i.e.…”

“…the theorem of Lebowitz and Costin [13] ensures a Gaussian behaviour for the scaled particle number fluctuations as the volume grows larger for a large class of determinantal processes.…”

Definitions and Basic Properties

“…Heuristical derivations of Gaussian limit for linear statistics may be found in Politzer [95], Beenakker [24,25], Costin, Lebowitz [36] while some of the references where rigorous analysis is performed include Johansson [71], Basor [22], Sinai, Soshnikov [100], Soshnikov [103], Bai, Silverstein [13] and Dumitriu, Edelman [43]. It should be noticed that similar results are obtained for spectrum fluctuations of matrices from classical compact groups, but they will be discussed in detail in Chapter 5.…”