Critical and umbilical points of a non-Gaussian random field

Random fields in nature often have, to a good approximation, Gaussian characteristics. For such fields, the number of maxima and minima are the same. Furthermore, the relative densities of umbilical points, topological defects which can be classified into three types, have certain fixed values. Phenomena described by nonlinear laws can, however, give rise to a non-Gaussian contribution, causing a deviation from these universal values. We consider a random surface, whose height is given by a nonlinear function of a Gaussian field. We find that, as a result of the non-Gaussianity, the density of maxima and minima no longer match and we calculate the relative imbalance between the two. We also calculate the change in the relative density of umbilics. This allows us not only to detect a perturbation, but to determine its size as well. This geometric approach offers an independent way of detecting non-Gaussianity, which even works in cases where the field itself can not be probed directly.

Figure 1

A realization of a Gaussian field with periodic boundary conditions.

Figure 2

Histograms of the values of ${10}^6$ minima obtained from simulations, together with the distribution given by Eq. (37), for (a) a disk spectrum ($\lambda =\frac{3}{4}$); (b) a Gaussian spectrum ($\lambda =\frac{1}{2}$).

Figure 3

Histogram of the values of ${10}^6$ minima obtained from simulations, together with the distribution given by Eq. (37), for a ring spectrum ($\lambda =1$). No minima with a positive value of $H$ were found.

Figure 4

The skewness (${\gamma }_1$) and kurtosis (${\gamma }_2$) of the distribution [Eq. (37)] as a function of $\lambda$ [see Eqs. (45) and (46)].

Figure 5

$\Delta n$ for $h=H+\varepsilon H^2$ as a function of $\varepsilon$, where $H$ has a disk spectrum ($\lambda =\frac{3}{4}$). The data points stem from simulations, the solid curve is Eq. (47). The two graphs are for different ranges of $\varepsilon$.

Figure 6

$\Delta n$ for $h=H+\varepsilon H^2$ as a function of $\varepsilon$, where $H$ has a Gaussian spectrum ($\lambda =\frac{1}{2}$). The data points stem from simulations, the solid curve is Eq. (47). The two graphs are for different ranges of $\varepsilon$.

Figure 7

One set of curvature lines (black) around (a) a lemon, (b) a monstar, and (c) a star. The other set shows the same pattern in all cases. The red circle and line segments show how the principal direction rotates around the umbilical point.

Figure 8

The monstar fraction ${\alpha }_M$ of $H+\varepsilon H^3$ as a function of $\varepsilon$, where $H$ has a disk spectrum ($\mu =\frac{16}{27}$). The data points stem from simulations, the solid line is Eq. (69).

Figure 9

The monstar fraction ${\alpha }_M$ of $H+\varepsilon H^3$ as a function of $\varepsilon$, where $H$ has a Gaussian spectrum ($\mu =\frac{2}{9}$). The data points stem from simulations, the solid line is Eq. (69).

Figure 10

Identifying a critical point. The four black squares are grid points at which $H_x$ and $H_y$ are known. At the red (blue) dots, $H_x=0$ ($H_y=0$) under the assumption that $H_x$ ($H_y$) is linear between the two grid points. The two contour lines $H_x=0$ and $H_y=0$ then intersect inside the square, indicating the existence of a critical point.