We consider fluctuations in the distribution of critical points - saddle points, minima and maxima - of random gaussian fields. We calculate the asymptotic limits of the two point correlation function for various critical point densities, for both long and short range. We perform the calculation for any dimension of the field, provide explicit formulae for two and three dimensions, and verify our results with numerical calculations.
Random gaussian fields constitute an important paradigm in physics. They serve as a simplified model for diverse physical ingredients such as the energy landscape of glassy systems [1,2,3,4], wave functions of quantum systems with chaotic classical dynamics [5], and the fluctuations of the Cosmic Microwave Background radiation [6]. The critical points of these gaussian fields -minima, maxima and saddle points -characterize the behavior of the entire field and thus carry a large amount of information. For instance, large scale structure in the universe can be analyzed using the critical-point density [7]. In other cases critical points are directly related to physical phenomena such as reflection patterns of sunlight from the sea surface [8], and the onset of glassy behavior in a complex energy landscape, see [1] and references therein. An additional novel context where critical points play an important role is that of cold atoms in disordered optical potentials [9,10,11]. In these systems, proper detuning of the light frequency forces the atoms to gravitate to the local extremum points of the intensity, so that statistics of the critical points map directly to statistics of the atom positions.
Most studies of critical points have focused on their density. However, in order to characterize their statistical properties it is also important to understand their correlation functions. In particular, in order to calculate fluctuations of the number of critical points, within a given volume of the system, one needs the two-point correlation function.
In this work we shall address the problem of pair correlations of critical points. The one-dimensional version of this problem was solved many years ago [12,13]. In two dimensions, some asymptotic expressions are known [14,15], along with some numerical data [16] (see also [17,18]). For a comprehensive review, concentrating on wave systems, see [19] and references therein. We shall provide a detailed derivation for the general asymptotic formula of critical point correlations in any dimension d ≥ 2, for both long and short distances. Some of this work, especially in the longdistance limit, may be seen as an expansion and generalization of methods developed previously, see e.g. [20,21,14].
There are several types of densities that can be associated with critical points. In order to present them, let φ(r) be a gaussian random field in d spatial dimensions, r = (r 1 , r 2 • • • r d ), with zero mean and an isotopic correlation function
We shall assume that the second derivative of G(r) is finite at r = 0 and vanishes at r → ∞. We note that in general this implies that r -γ G(r) = 0 vanishes in the limit r → ∞, for 0 ≤ γ < 2.
Critical points are the points where the gradient of the field vanishes, and their density is given by the well known Kac-Rice formula:
where H is the Hessian matrix whose elements are the second derivatives of the field:
This density, which we shall refer to as the unsigned density of critical points gives an equal weight and sign to each one of the critical points. There are, however, many other possible choices of density. For instance one may define the signed density as in Eq. ( 2) but omitting the absolute value operation. In two spatial dimensions this density assigns a plus sign to minima and maxima points and a minus sign to saddle points. Another possibility is to define a density of only minima points of the gaussian field. The main focus of this work is the correlation functions of the unsigned density and the density of minima points:
where ρ(r) denotes one of these critical-points densities. However, the methodology we shall develop in order to calculate these functions can be generalized in a straightforward manner to the correlation functions of different types of critical point densities.
The central result of this work is that the asymptotic behavior of the correlation function is
where ρ is the average density of critical points, α 1 , α 2 and α 3 are constants which depend on the type of the critical-point density and the dimensionality of the system d,
is the Hessian matrix of the correlation function of the gaussian field, and σ is its typical length scale. In this formula k = 0 corresponds to the correlation of the unsigned density, while k = 3 corresponds to the correlation of mimina points. Notice that the correlation function also contains a δ-function at the origin, C (r → 0) = ρ δ (r), due to trivial self correlations of the critical points. The paper is arranged as follows: In Sec. 2 we shall sketch the derivation of asymptotic form of the correlation function (4) deferring details to Appendix A. Then we shall apply these results to find the correlation functions in two and three dimensions, in the long-range asymptotic limit. Finally, we will discuss some examples and conclusions in Sec. 4. Appendix A contains the technical details of the derivation, and Appendix B provides information regarding the numerical
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