A plane spends the same reach on a disc

Four essays of this argument have computed properties of a defect band from three ingredients: how often a run of low sites occurs, how strongly two of them couple across the sites between, and how closely the levels around them are spaced. The share of runs in a resonant pair is built out of those three and nothing else, and it takes between eight and a hundred and twenty decades of chain length to go from nothing to a half depending on the run.

Every one of those three ingredients is dimension-independent. A run of low sites occurs at a rate fixed by the concentration. Two runs couple by an amount falling exponentially with the distance between them, because the state a run holds decays exponentially into the host. The levels of an N-site system are N times a density spread over a fixed energy range, so their spacing falls as 1/N. Nothing in any of the three mentions how many directions there are, and the whole of the coupling distribution the resonance criterion is applied to is built from them.

The one-dimensional part is somewhere else, and it is a counting statement. How many candidate twins lie within a separation s? In a chain there are s of them, one per site along the line. On a plane there are πs², one per site of a disc.

That is the whole of the difference, and it is worth thirty-seven orders of magnitude.

What the object is on a plane #

A run of six low sites is a one-dimensional object and it has to be placed in the plane in some definite way, so the choice is stated rather than left implicit. The runs here are horizontal runs of exactly six low sites in the rows of a square net — six consecutive low sites in a row, bounded by a high site at each end — with every site low independently with probability one half.

That is the same object the chain calculation had, put in a plane, and it is the right object for a layered or anisotropic material where a defect extends along one crystallographic direction. It is not the only choice available: a two-dimensional cluster of six low sites is a different object with a different density and a different level, and the calculation for it needs a cluster census rather than a run census. What this essay isolates is the neighbour count, so it changes the neighbour count and nothing else.

The density is then unchanged. Six lows bounded by two highs occur at $x^{6}(1-x)^{2}$ per site whether the rows are part of a chain or of a net, and the nets say so: three nets of five hundred by five hundred sites hold 2,923 runs of six, a density 0.998 of the closed form.

A run touching an edge is discarded rather than counted, which is the one bookkeeping decision in the census and is made the way the chain census made it. A run cut off by a boundary is not a run of six; it is a run of at least six, and counting it as six would inflate the density by the perimeter’s share of the net. At five hundred sites a side that share is under a per cent, which is why the measured density agrees to two parts in a thousand and why the same decision was invisible in the chain.

The law, measured rather than written down #

The chance that no twin lies within s is the chance that none of the candidate sites in that region holds one. In a chain that is (1 − ρ)^s. On a plane it is (1 − ρ) raised to the number of sites in a disc of radius s, which for a small density is exp(−ρπs²).

That is a formula and it is easy to write down wrongly, so it is measured. For every run of six in the nets, the distance to its nearest twin is computed, and the cumulative distribution of those distances is compared against the closed form with nothing fitted.

Beyond a separation of six sites the two agree to a few points and at sixteen sites to a tenth of one. That is the range the half-way separation falls in, and it is the range the argument uses.

The comparison is made against the distance to the nearest twin rather than against a count of twins within a distance, and the two are the same statement: the share with at least one inside s is one minus the share with none. Measuring the nearest is cheaper and it is also the quantity the resonance criterion is about, since a run in a resonant pair needs one partner and not a number of them. Where the two would come apart is at large s, where a run has several partners and the criterion would have to say what a triple does — which is the question a count rather than an average was already skirting, and is not reached here because the half-way separation is where a typical run has exactly one.

Where the law is wrong, and what it forgets #

Below a separation of about six sites the closed form over-predicts, and the size of the discrepancy is worth reporting because the cause is not subtle.

At s = 2 the nets say 2.9 per cent of runs have a twin that close and the formula says 4.8. At s = 4 they say 12.9 against 17.8. At s = 8, 50.2 against 54.4. At s = 16, 95.6 against 95.7.

A run of six is not a point, and two of them cannot overlap. In the same row their centres cannot be closer than seven sites; in adjacent rows they can be closer, but the region a twin is excluded from is still several sites across. A Poisson process in the plane places points with no regard to each other and therefore allows configurations that cannot occur — so it over-counts close pairs, which is exactly the direction observed.

The measured discrepancy is a hard-core correction and its size is the object’s size. It matters here because the half-way separation on a plane is 7.5 sites, right in the region where the correction is still worth four points: the measured half-way separation is nearer 8.0, which moves the half-way length up by about a tenth of a decade. In a chain this correction is invisible, because the half-way separation there is 177 sites and a hard core seven sites across is four per cent of it.

That is the first thing dimension does to this argument that is not the exponent. Shortening every distance by a factor of twenty brings the objects’ own size into view.

Thirty-seven orders of magnitude #

The half-way separation is where the exponent reaches ln 2, so it is (ln 2 / ρc)^(1/d) with c the shape constant — one for a line, π for a disc, 4π/3 for a ball. The 1/d is the whole finding.

At half filling the half-way separation is 177.45 sites in a chain, 7.52 in a plane and 3.49 in a solid. The reach a system buys grows by 4.61 sites per decade of size, so those separations are reached at 10^40.78, 10^3.93 and 10^3.06 sites.

A chain needs a rocky asteroid’s worth of atoms and a plane needs a patch ninety-two sites square.

The patch is small enough to print, which is the most convincing form the result takes. It holds thirty-five runs of six; eighteen of them — fifty-one per cent — have another within the half-way separation. The closed form said fifty, from a formula with a π in it and nothing fitted, and the drawn net agrees.

There is no partly resonant plane #

The half-way length is not the most useful number to come out of this. The window is.

The coupling reaches nothing until the level spacing has fallen far enough, which happens at 200 sites for runs of six at half filling, in every dimension — the level spacing does not know the dimension either. From there to the half-way point is 38.5 decades in a chain, 1.63 in a plane and 0.76 in a solid.

Under two decades is not a slide, it is a threshold. It means the partly resonant regime that the chain occupies for thirty-eight decades — and that every real one-dimensional sample is therefore in — does not exist on a plane in any practical sense. A two-dimensional sample either has too few sites to hold a statistically meaningful number of runs at all, or its runs of that length are in resonant pairs.

So the whole apparatus of the essays before it, the per-decade rise and the saturation and the boundary of what matter reaches, is a one-dimensional apparatus. It answers a question that two- and three-dimensional materials do not pose.

The same arithmetic run at other run lengths says how far that extends, and the answer is: everywhere. In a plane, runs of four reach half at 10^2.87 sites and runs of eight at 10^5.28 — so the whole argument of run lengths that spans twenty-five decades in a chain is compressed into two and a half in a plane. The diagonal boundary across run length and concentration, which was the previous essay’s finding, has no plane analogue at all: every combination is on the saturated side of it.

Which way the conclusion cuts #

It would be easy to read this as the earlier essays being a detour, and it is closer to the opposite.

The reason the chain shows a slow rise at all is that the exponent grows only as fast as the reach, and the reach grows as the logarithm of the size. Raising a logarithm to a power is the cheapest way there is to make something grow, and even that is enough: a square turns forty decades into four. So the slow rise was never robust, and the essays that found it were measuring the most fragile arrangement of the three — which is worth knowing about a result, and is the kind of thing only a dimension sweep says.

And the one-dimensional case is the physically interesting one for exactly that reason. A quantity that slides over thirty-eight decades is a quantity a sample’s own size controls; a quantity with a 1.6-decade threshold is a quantity the material controls. The first is a measurement problem and the second is a material property, and this argument has been studying the measurement problem.

What the three dimensions share, and what is assumed #

The shape constants are geometric and are not fitted. A disc of radius s holds πs² sites of a square net to within its perimeter, and a ball holds 4πs³/3. Those are the only numbers that differ between the three calculations, and the refusal this figure runs is the check that saying so is true: a plane whose neighbour count is taken as linear must reproduce the chain’s own answer exactly, since nothing else differs between them.

The coupling law is the chain’s fit, used in the plane. ξ and the prefactor come from two runs in a chain at a range of separations, and using them in a plane assumes the coupling between two runs depends on the distance between them and not on the direction. In a square net with hopping along rows and columns that is false in detail — a twin directly above is reached by a different path than one along the row — and the correction is a factor of order one in the prefactor, which moves the half-way separation as its logarithm and the half-way length by a fraction of a decade. The thirty-seven decades do not depend on it.

The level spacing is the chain’s. The gap levels of a plane’s runs occupy the same energy range, because they are the same runs, and their number is the same per site. That is why the spacing is unchanged and it is why the onset length is the same in all three dimensions.

The runs of a plane are still one-dimensional objects. The level a run holds comes from treating it as a particle in a box of six sites, and in a plane a run’s low sites have neighbours above and below them as well as along the row. Those neighbours are low half the time, so a run of six in a net is generally attached to more low sites than a run of six in a chain is, and its level is not the chain’s box level. That is the largest single approximation in this essay and it is the reason the object was chosen rather than defended: what is isolated here is the neighbour count, and the level is held fixed so that the neighbour count is the only thing that moved.

And nothing here is a percolation calculation. At half filling a square net is above its site percolation threshold and its low sites form a spanning cluster, so the picture of isolated runs separated by host is not what half a net looks like — the runs of exactly six counted here are the rare configurations where a row’s low sites happen to be bounded, and they are embedded in a connected low region that the level calculation ignores. The count is right; the tight-binding picture behind it is a chain’s.

What a plane would look like instead #

If the partly resonant regime does not exist in two dimensions, the question a plane poses is a different one, and it is worth naming because the essays before it have all the apparatus for it.

A plane’s runs of a given length are resonant with each other essentially always, so the interesting quantity is not what share of them are paired but how wide the band they make is. That is a question about the distribution of couplings rather than about a threshold on it, and the distribution is the object an earlier essay built: a width is a mean or a second moment of it, where a resonant share is a count above a cutoff. The two readings of the same distribution are the two regimes, and which one a material is in is decided by the dimension and not by anything a sample can be made to do.

That is also why the level spacing kept turning up as the yardstick. A spacing is the right comparison when the question is whether two levels are distinguishable, which is the question in a chain because most pairs are not coupled. When most pairs are coupled the spacing is not a yardstick for anything; the band’s own width is. The apparatus this argument built is a threshold apparatus and the plane needs a moment apparatus, and nothing about the change of dimension announces which one applies.

Dimension is in the counting, not in the physics #

The habit: when a result is derived for one dimension, find which step of it mentions the dimension.

Here three steps did not and one did, and the one that did is a line of combinatorics rather than a piece of physics — the number of sites within a distance. It is the least interesting line in the derivation and it carries the entire size dependence, because it sits in an exponent.

The corollary is about where to look for fragility. A result’s sensitivity is not spread evenly through its derivation; it concentrates wherever a quantity enters an exponent. Both of this argument’s large numbers — the hundred and seventy decades of coupling and the forty decades of length — come from exponentials, and both turn out to be movable by changes that look like bookkeeping. A factor of π in a count of neighbours moved an answer by thirty-seven orders of magnitude, and nothing about the π is deep.

Who counted what, and when #

The Poisson approximation for nearest-neighbour distances in a random point pattern, and the exp(−ρπs²) form in two dimensions, are standard spatial statistics and long predate any of this. The hard-core correction to it is the classical difference between a Poisson process and a Matérn process. The exponential decay of a defect state’s coupling and the criterion that a set of levels is a band when the coupling exceeds the spacing are standard tight-binding, and are the ones the essays before it used.

What is computed here is the run census and the nearest-twin distribution in generated square nets, the hard-core discrepancy against the closed form, and the half-way length and transition window in one, two and three dimensions from the same coupling and spacing.

The number worth carrying is 1/d — an exponent on a separation, which is the only place the dimension enters and is worth thirty-seven decades.

Still open: a cluster rather than a run, and the direction of the coupling #

The obvious open question is the object. A horizontal run of six in a plane is a defensible object and it is not the natural one: a two-dimensional material’s defects are clusters, and a cluster of six sites has a level that depends on its shape as well as its size, so a census of shapes would replace the single box level with a distribution of them. That changes the argument qualitatively rather than quantitatively, because runs of one length are levels at one energy and clusters of one size are not — the resonance condition would then involve a detuning, and two clusters of the same size and different shapes would be off resonance with each other. Whether a plane’s defect band is made of resonant pairs or of detuned ones is a question this essay cannot ask and the shape census can.

The nearer question is the direction of the coupling. The exponential law used here is isotropic, and on a square net with equal hopping along both directions the coupling between two runs is not: a twin in the same row is connected along the row, a twin directly above is connected by a path through the sites between, and the two have different prefactors at the same distance. Computing the splitting for the two orientations at a few separations would give the anisotropy, and since the reach enters as a logarithm, an anisotropy of a factor of two is worth a fraction of a decade — small, and the only way to know it is small is to compute it.