The pieces no formula counts

Pieces minus holes on a random surface cut a random landscape at a height $u$ and counted what was above the cut. The region above the cut breaks into pieces with holes in them, and while neither count has a formula, their difference does: the average Euler characteristic, pieces minus holes, is a constant times $u$ times the bell curve at $u$. The essay ended on what that leaves out. The pieces themselves have no formula, and for the simplest random vibration — waves of a single wavelength — the number of pieces at the middle height was known to grow in proportion to the area, with a constant nobody knows.

This essay counts them. The object is a random plane wave: a sum of many plane waves with the same wavelength, travelling in random directions with random phases, the standard model for a high-frequency vibration of a drumhead or the wavefunction of a chaotic quantum billiard. At the middle height, $u = 0$, the regions where the wave is positive and where it is negative are called its nodal domains, and the curves between them its nodal lines.

The picture shows what makes the count hard. The positive region is a tangle of blobs and long branching corridors, some of them crossing most of the square, some no bigger than a wavelength. Whether two blobs are one piece or two is decided at the narrow places where they nearly touch, and at each such place the wave has a saddle point: the surface rises towards two blobs and falls towards two gaps, and the sign of the wave at the saddle decides whether the blobs join or the gaps join. Every saddle is a coin that decides one connection, and the number of pieces is the result of all of them together — a global tally that no inspection of one neighbourhood at a time can reproduce.

Why a random wave #

The random plane wave is not an arbitrary choice of random surface. Michael Berry proposed in 1977 that the high-frequency vibrations of a drum whose shape makes the motion of a billiard ball inside it chaotic — a stadium, say, rather than a circle — look locally like random superpositions of plane waves of one wavelength. A billiard in a rectangle bounces in a perfectly regular way, as a bounce is a fold of the table showed by unfolding the table into a tiling of the plane; in a stadium the bounces scatter, and the vibration’s waves arrive from every direction with phases that look random. Berry’s conjecture is unproved but extensively confirmed numerically, and it is why the nodal domains of a random wave are taken as the model for the nodal domains of a chaotic drum’s high modes.

The wavelength sets the scale. A drum’s $n$-th vibration mode has, by Weyl’s law, a spatial frequency $k$ with $\tfrac{k^2}{4\pi} \times \text{area} \approx n$, so the unit used here — area times $k^2/4\pi$ — is just the mode number. Counting nodal domains per unit is counting them per mode: a random wave of the size of the $n$-th mode of a drum has about $0.06\,n$ nodal domains.

What the Euler characteristic knows #

The Euler characteristic of the positive region is the number of pieces minus the number of holes, and its average is known exactly. For a random field with the smoothness of a plane wave, the formula of the previous essay gives an average of $u\,\varphi(u)$ per unit of area-times-energy — with $\varphi$ the bell curve and the unit, here and throughout, the area of the region multiplied by $k^2/4\pi$, where $k$ is the wave’s spatial frequency. At $u = 0$ that is nought: on average there are exactly as many holes as pieces.

The measured difference follows the formula at every level drawn. The two counts it is the difference of behave quite differently. At high levels the region above the cut is a scatter of islands with no holes, and the pieces nearly equal the Euler characteristic. At low levels the region is almost everything, a single sheet with holes in it, and the holes nearly equal it with the sign reversed. At $u = 0$ both counts are at their largest, close to 0.03 per unit each, and they cancel. The formula, which is an integral of local quantities, knows their difference precisely and nothing at all about either. Beyond about $|u| = 1.25$ the islands that appear just above the cut are smaller than the grid can resolve, and the computation stops there rather than report a number it cannot see.

What counts as a hole needs a word. A hole of the positive region is a bounded piece of the negative region, sitting inside it — a loop in the positive region that goes round something not in it, which is the sense a hole is a cycle that bounds nothing made precise. In the plane the two descriptions agree, so the holes of the positive region are exactly the negative pieces it encloses, and at $u = 0$ the positive and negative pieces are the same kind of object, which is why their numbers balance.

Counting targets by their holes used the same additivity to count targets from sensor readings without ever identifying a target, and it worked because the targets had no holes. The nodal domains do have holes, as many as pieces at the middle level, and that is exactly why their number escapes the additive quantity.

Counting the pieces #

To count the pieces, each wave was evaluated on a fine grid, sixteen points to a wavelength, and the grid points of each sign were joined into connected pieces. The difficult cells are the ones whose four corners alternate in sign — two diagonally opposite corners positive, the other two negative — because the grid alone cannot say which diagonal is connected. A saddle lies inside such a cell, and the computation evaluates the wave exactly at the cell’s centre and joins the diagonal that has the centre’s sign. Every saddle is decided by the wave itself.

The other difficulty is the edge of the window. A piece that touches the edge may continue outside it. Counting such pieces overcounts, since the edge has cut one piece into several; discarding them undercounts. Both errors are proportional to the window’s perimeter, so both counts were made in windows of six sizes and extrapolated to a window with no edge.

The two estimates come from opposite sides and meet at the same value, 0.0599, with a standard error of about 0.0005 from the scatter of the 51 fields about the straight lines. That is the number of nodal domains of a random plane wave per unit of area times $k^2/4\pi$ — equivalently, roughly one nodal domain for every seventeen units.

Far below the bound #

How many nodal domains can a vibration have? Richard Courant proved in 1923 that the $n$-th mode of any drum has at most $n$. In one dimension the bound is attained by every mode: the $n$-th mode of a string is a sine with $n$ humps, each a nodal domain, as the modes of a plucked string keeps its corners are. In two dimensions Åke Pleijel showed in 1956 that the bound is never attained for large $n$: asymptotically at most about $0.69\,n$, a constant that comes from the Faber–Krahn inequality: no domain can vibrate at the given frequency unless its area is at least that of a disc of the matching size. Random waves sit far below even that, at about $0.06\,n$ — a tenth of what is possible. Most of the domains a vibration could have, it does not have, because the nodal lines of a random wave do not cut the surface into the small, round, regular cells that the bound requires; they wander, merge at saddles and enclose large tangled regions.

The prediction from percolation #

Fedor Nazarov and Mikhail Sodin proved in 2009 that the number of nodal domains of a random plane wave in a large region, divided by the region’s area, converges to a constant, and that the count in a single large region is almost certainly close to it. They did not compute the constant, and no one has.

The best prediction is Eugene Bogomolny and Charles Schmit’s, from 2002. Their idea was that the nodal lines of a random wave behave like the boundaries of clusters in critical percolation — the model the moment a giant appears met for random graphs, here on a lattice in the plane, at the threshold where clusters of all sizes coexist. The level $u = 0$ is critical for a reason of symmetry. The positive and negative regions of a random wave are statistically identical — the wave and its negative are equally likely — so neither can be the one that spans the plane, and both sit exactly at the threshold where an infinite piece would form. It is the same reasoning that places bond percolation on the square lattice at its threshold when each bond is open with probability one half, Harry Kesten’s theorem of 1980; sharp, or merely a threshold described the window in which such a transition happens. At the threshold, clusters of every size appear, and the nodal domains of the hero figure, from tiny blobs to corridors crossing the window, are that mixture of scales.

In their model the saddle points are the bonds of a percolation lattice, each one open or closed at random with even odds, and counting clusters of critical bond percolation exactly, a result from statistical physics, gives

$$\frac{3\sqrt3 - 5}{\pi} \approx 0.0624$$

nodal domains per unit. The number is built from two ingredients. The saddle points of a random wave have a density that can be computed exactly from the wave’s covariance, as averages of local quantities always can; Bogomolny and Schmit take the saddles as the sites of a square lattice, joined in the way a saddle joins two blobs or two gaps. And the number of clusters per site in critical bond percolation on the square lattice is known exactly, $(3\sqrt3 - 5)/2$, a classical result of statistical physics derived from the lattice’s hidden algebraic structure. Multiplying a local density by a percolation count is what gives a closed form — and the measured value shows that the local density and the percolation count are not quite the right things to multiply.

The measured 0.0599 is five standard errors below it. Careful published computations agree: the most precise give about 0.0589, and the difference from 0.0599 here is within what the extrapolation’s systematic uncertainties allow. The percolation model is close — within five per cent — and it is not right.

Where the model is right #

The model’s failure is in the count. In shape, the domains do behave like percolation clusters.

Large domains are rare, and how rare follows a power law: the number of domains with area near $A$ falls like $A^{-\tau}$. For critical percolation in the plane the exponent is known exactly, $\tau = 187/91 \approx 2.055$, and the large nodal domains fall off with a fitted slope of 2.15, within the uncertainty of a fit over this range. The large-scale structure of the nodal lines is percolation’s, and that part of Bogomolny and Schmit’s picture has been tested from many directions and holds up. The count is decided mostly by the small domains, a wavelength or two across, where the wave’s smoothness and the correlations between nearby saddles matter, and the model, which treats the saddles as independent, gets those slightly wrong. A saddle is not an independent coin: whether a saddle opens is correlated with the saddles next to it, because the same few waves shape them all.

Every saddle is a decision #

How much the saddles matter shows most clearly when they are decided wrongly.

On a coarse grid there are many ambiguous cells — over a thousand per field here — and the rule used to settle them changes the answer. A rule that always joins the positive diagonal merges positive pieces and splits negative ones; the opposite rule does the reverse. Either way the total moves only a little, but the balance between the two signs is wrong, and the Euler characteristic at $u = 0$ says it must be even on average. Only when every saddle is decided by the wave itself do positive and negative pieces come out equal, 0.0318 and 0.0319. The number of pieces is not a sum of local quantities; it is a function of the joint outcome of every saddle, and that is the precise sense in which it has no formula of the kind the Euler characteristic has.

One window is nearly enough #

Nazarov and Sodin’s theorem has a second half, about how much a single count varies.

The counts for different fields of the same size cluster tightly, and the clusters tighten as the windows grow. The number of domains in a large region is not only proportional to the area on average; for a single random wave it is close to the average with overwhelming probability. That concentration is what makes the measurement possible at all: 51 fields are enough to pin the constant to a few parts in a thousand, because each field already averages over hundreds of domains. The drift of the clusters downward with size is the edge effect, the same one the extrapolation removes.

Still open: the constant #

The Nazarov–Sodin constant for random plane waves is not known in closed form. Its value is about 0.059 per unit, measured many ways; Bogomolny and Schmit’s percolation value, 0.0624, is close and wrong; and whether the true constant has any closed expression at all is open. Rigorous bounds are far apart: the best proven lower bounds are much smaller than the measured value, because proving that domains exist in quantity requires controlling the saddles’ correlations, which is exactly what the percolation model leaves out.

Related constants are open too. The number of nodal domains of a single eigenfunction of a drum, rather than of a random wave, is conjectured to follow the same law for drums whose classical dynamics are chaotic, and that conjecture — which ties the counting of pieces to the question of what a random surface’s Euler characteristic averages and to quantum chaos — has been tested numerically and is not proved for any drum. What would settle the matter is unclear. A closed form would presumably come from a description of the saddles’ correlations precise enough to correct the percolation picture, and no such description is in sight; a proof that no closed form exists is not even well posed. Numerically the constant could be pinned down further — the uncertainty here is a few parts in a thousand, and the larger published computations, run on far bigger grids and many more waves, are more precise still — but more digits of a number with no known formula do not by themselves suggest one.

For random fields with other spectra, Nazarov and Sodin’s theorem gives a constant for each, and how the constant depends on the spectrum is understood only in examples.

The pieces and their difference #

The region where a random vibration bulges up is a tangle of pieces with holes in them. Their difference has an exact average, $u\,\varphi(u)$ per unit at height $u$, confirmed level by level. Their number has none: at the middle height it is about 0.0599 per unit by direct count of 51 random waves, below percolation’s prediction of 0.0624, while the sizes of the pieces follow percolation’s law. The count is a global quantity, the outcome of every saddle at once, and the one exact formula in the subject sees only the part of it that the saddles cancel.