The Spectrum of Riemannium
The Spectrum of Interstatium
How things distribute themselves in space or time or along some more abstract dimension is a question that comes up in all the sciences. An astronomer wants to know how galaxies are scattered around the universe; a biologist might study the distribution of genes along a strand of chromatin; a seismologist records the temporal pattern of earthquakes; a mathematician ponders the sprinkling of prime numbers among the integers. Here I shall consider only discrete, one-dimensional distributions, where the positions of items can be plotted along a line.
Figure 1 shows samples of several such distributions, some of them mathematically defined and others derived from measurements or observations. All of the samples have been scaled so that exactly 100 levels fit in the space allotted. Thus the mean distance between levels is the same in all cases, but the patterns are nonetheless quite diverse. For example, the earthquake series is highly clustered, which surely reflects some geophysical mechanism. The lower-frequency fluctuations of tree-ring data probably have both biological and climatological causes. And it's anyone's guess how to explain the locations of bridges recorded while driving along a stretch of Interstate highway.
In analyzing patterns of this kind, there is seldom much hope of predicting the positions of individual elements in a series. The aim is statistical understanding—a description of a typical pattern rather than a specific one. I shall focus on two statistical measures: nearest-neighbor spacings and the two-point correlation function.
The simplest of all distributions is a periodic one. Think of a picket fence or the monotonous ticking of a clock: All the intervals between elements of the series are exactly the same. The obvious counterpoint to such a repetitive pattern is a totally random one. And between these extremes of order and disorder there are various intermediate possibilities, such as a "jiggled" picket fence, where periodic levels have been randomly displaced by a small amount.
A graph of nearest-neighbor spacings readily distinguishes among the periodic, random and jiggled patterns (see Figure 2). For the periodic distribution, the graph is a single point: All the spacings are the same. The nearest-neighbor spectrum of the random distribution is more interesting: The frequency of any spacing x is proportional to e-x. This negative-exponential law implies that the smallest spacing between levels is the likeliest. The jiggled pattern yields a bell-shaped curve, suggesting that the nearest-neighbor intervals have a Gaussian distribution.
The pair-correlation function mentioned by Montgomery and Dyson captures some of the same information as the nearest-neighbor spectrum, but it is calculated differently. For each distance x, the correlation function counts how many pairs of levels are separated by x, whether or not those levels are nearest neighbors. The pair-correlation function for a random distribution is flat, since all intervals are equally likely. As the distribution becomes more orderly, the pair-correlation function develops humps and ripples; for the periodic distribution it is a series of sharp spikes.