Fluctuation and entropy in spectrally constrained random fields

Abstract

We investigate the statistical properties of translation invariant random fields (including point processes) on Euclidean spaces (or lattices) under constraints on their spectrum or structure function. An important class of models that motivate our study are hyperuniform and stealthy hyperuniform systems, which are characterised by the vanishing of the structure function at the origin (resp., vanishing in a neighbourhood of the origin). We show that many key features of two classical statistical mechanical measures of randomness—namely, fluctuations and entropy, are governed only by some particular local aspects of their structure function. We obtain exponents for the fluctuations of the local mass in domains of growing size, and show that spatial geometric considerations play an important role—both the shape of the domain and the mode of spectral decay. In doing so, we unveil intriguing oscillatory behaviour of spatial correlations of local masses in adjacent box domains. We describe very general conditions under which we show that the field of local masses exhibit Gaussian asymptotics, with an explicitly described limit. We further demonstrate that stealthy hyperuniform systems with joint densities exhibit degeneracy in their asymptotic entropy per site. In fact, our analysis shows that entropic degeneracy sets in under much milder conditions than stealthiness, as soon as the structure function fails to be logarithmically integrable.

Appendix

For the sake of completeness, the Riemann-Lebesgue lemma is stated below, see [Eps08, Lemma 4.2.1].

Lemma 4

(Riemann Lebesgue lemma). If f is $L^1$ integrable on ${\mathbb {R}}^d$, that is to say, if the Lebesgue integral of |f| is finite, then the Fourier transform of f satisfies

$$\begin{aligned} \hat{f}(z):=\int _{{\mathbb {R}}^d} f(x) \exp (-iz \cdot x)\,dx \rightarrow 0\text { as } |z|\rightarrow \infty . \end{aligned}$$

Theorem 9

([Kat04, P. 152]). If $e^{P(\xi )}$ is the Fourier-Stieltjes transform of a positive measure, with P a polynomial, then $\deg P\le 2$.

For the sake of completeness, the proof of Lemma 3 is given below.

Proof of Lemma 3

Let g be the joint density function of the random vector G. Then g is given by

$$\begin{aligned} g(x)=\frac{1}{\det (\sqrt{2\pi \Sigma _d})}e^{-\frac{1}{2}x^t\Sigma _d^{-1} x}, \end{aligned}$$

where $\Sigma _d=(\sigma (i,j))_{d\times d}$ with $\sigma (i,j)=\text{ E }[X_iX_j]$ and $x=(x_1,\ldots , x_d)$. Let f be the continuous density function of the random variable X. The relative entropy (also known as Kullback-Leibler divergence) between f and g is given by

$$\begin{aligned} 0\le D_{KL }(f\Vert g)=\int f(x)\log (\frac{f(x)}{g(x)})dx=-h(X)-\int f(x)\log (g(x))dx. \end{aligned}$$

Note that we have

$$\begin{aligned} \log (g(x))=-\frac{1}{2}\log \det (2\pi \Sigma _d)-\frac{1}{2}x^t\Sigma _d^{-1} x. \end{aligned}$$

Which implies that, as $\text{ E }[X_iX_j]=\text{ E }[G_iG_j]$ for all i, j,

$$\begin{aligned} \int _{-\infty }^\infty f(x)\log (g(x))dx&=-\frac{1}{2}\log \det (2\pi \Sigma _d)-\frac{1}{2}\int f(x)(x^t\Sigma _d^{-1} x)dx \\&=-\frac{1}{2}\log \det (2\pi \Sigma _d)-\frac{1}{2}\int g(x)(x^t\Sigma _d^{-1} x)dx=-h(G). \end{aligned}$$

Thus we have $h(G)-h(X)\ge 0.$ Moreover, the properties of Kullback-Leibler divergence imply that $h(X)=h(G)$ when $f=g$. Hence the result. $\quad \square$