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Register a new userFluctuations of the number of excursion sets of planar Gaussian fields
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For a smooth, stationary, planar Gaussian field, we consider the number of connected components of its excursion set (or level set) contained in a large square of area $R^2$. The mean number of components is known to be of order $R^2$ for generic fields and all levels. We show that for certain fields with positive spectral density near the origin (including the Bargmann-Fock field), and for certain levels $ell$, these random variables have fluctuations of order at least $R$, and hence variance of order at least $R^2$. In particular, this holds for excursion sets when $ell$ is in some neighbourhood of zero, and it holds for excursion/level sets when $ell$ is sufficiently large. We prove stronger fluctuation lower bounds of order $R^alpha$, $alpha in [1,2]$, in the case that the spectral density has a singularity at the origin. Finally, we show that the number of excursion/level sets for the Random Plane Wave at certain levels has fluctuations of order at least $R^{3/2}$, and hence variance of order at least~$R^3$. We expect that these bounds are of the correct order, at least for generic levels.
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Nazarov and Sodin have shown that the number of connected components of the nodal set of a planar Gaussian field in a ball of radius $R$, normalised by area, converges to a constant as $Rto infty $. This has been generalised to excursion/level sets at arbitrary levels, implying the existence of functionals $c_{ES}(ell )$ and $c_{LS}(ell )$ that encode the density of excursion/level set components at the level $ell $. We prove that these functionals are continuously differentiable for a wide class of fields. This follows from a more general result, which derives differentiability of the functionals from the decay of the probability of `four-arm events for the field conditioned to have a saddle point at the origin. For some fields, including the important special cases of the Random Plane Wave and the Bargmann-Fock field, we also derive stochastic monotonicity of the conditioned field, which allows us to deduce regions on which $c_{ES}(ell )$ and $c_{LS}(ell )$ are monotone.
We present a method, based on the correlation function of excursion sets above a given threshold, to test the Gaussianity of the CMB temperature fluctuations in the sky. In particular, this method can be applied to discriminate between standard inflationary scenarios and those producing non-Gaussianity such as topological defects. We have obtained the normalized correlation of excursion sets, including different levels of noise, for 2-point probability density functions constructed from the Gaussian, chi_n^2 and Laplace 1-point probability density functions in two different ways. Considering subdegree angular scales, we find that this method can distinguish between different distributions even if the corresponding marginal probability density functions and/or the radiation power spectra are the same.
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We study the behaviour of the point process of critical points of isotropic stationary Gaussian fields. We compute the main term in the asymptotic expansion of the two-point correlation function near the diagonal. Our main result implies that for a generic field the critical points neither repel nor attract each other. Our analysis also allows to study how the short-range behaviour of critical points depends on their index.
Insight into a number of interesting questions in cosmology can be obtained from the first crossing distributions of physically motivated barriers by random walks with correlated steps. We write the first crossing distribution as a formal series, ordered by the number of times a walk upcrosses the barrier. Since the fraction of walks with many upcrossings is negligible if the walk has not taken many steps, the leading order term in this series is the most relevant for understanding the massive objects of most interest in cosmology. This first term only requires knowledge of the bivariate distribution of the walk height and slope, and provides an excellent approximation to the first crossing distribution for all barriers and smoothing filters of current interest. We show that this simplicity survives when extending the approach to the case of non-Gaussian random fields. For non-Gaussian fields which are obtained by deterministic transformations of a Gaussian, the first crossing distribution is simply related to that for Gaussian walks crossing a suitably rescaled barrier. Our analysis shows that this is a useful way to think of the generic case as well. Although our study is motivated by the possibility that the primordial fluctuation field was non-Gaussian, our results are general. In particular, they do not assume the non-Gaussianity is small, so they may be viewed as the solution to an excursion set analysis of the late-time, nonlinear fluctuation field rather than the initial one. They are also useful for models in which the barrier height is determined by quantities other than the initial density, since most other physically motivated variables (such as the shear) are usually stochastic and non-Gaussian. We use the Lognormal transformation to illustrate some of our arguments.
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