The share that was read as a line

A disordered chain of sites, each low with probability x and high otherwise, traps electronic states in runs of consecutive low sites. A single run is a level in the gap; many runs of the same length are many copies of that level, and whether they form a band depends on how strongly neighbouring runs couple across the high sites between them. The coupling falls exponentially with the separation, so most pairs of runs are coupled by amounts a hundred and seventy decades apart, and the useful question is a count: how many runs have a neighbour close enough to couple more strongly than the spacing of the levels around them.

That count has a closed form. Runs of exactly L low sites occur at a rate ρ = x^L (1 − x)² per site, so the gap to a run’s nearest same-length neighbour is geometric, and the share of runs whose neighbour lies within a separation s is 1 − (1 − ρ)^s. The coupling falls to the level spacing at a separation s(N) = ξ ln($t_0$/spacing), and the spacing of an N-site chain falls as 1/N. For runs of six at x = 0.5 the share rises from 1.26 per cent at a thousand sites to 16.1 per cent at a million million.

What resonance means here is specific. Two runs of the same length hold levels at the same energy, and a coupling between them splits the pair into a bonding and an antibonding combination, the level between two defects doubled. If the splitting is smaller than the spacing of all the other gap levels nearby, the two runs still behave as two isolated levels that happen to be neighbours; if it is larger, they are one delocalised pair, and a band begins to be made of such pairs. The chain length at which the closest pair first qualifies was the onset of a band. The share asks the following question: of all the runs, how many have got there.

The share was read as a straight line in the logarithm of N, rising by ρξ ln 10 a decade, and on that reading the share never saturates and half of all runs of six would be resonant at a chain of about 10³¹ sites. The straight line came from approximating 1 − (1 − ρ)^s by ρs, which is accurate while ρs is small. The concentration of low sites moves ρ across four decades without touching ξ, so sweeping it is the way to see where that approximation holds and what the curve does where it does not.

Each decade adds what is left #

The separation at which a pair is resonant grows by exactly ξ ln 10 for every decade of chain length, because the spacing falls as 1/N and the separation is ξ times the logarithm of its reciprocal. The share is a function of that separation, so its rise per decade is the derivative of 1 − (1 − ρ)^s with respect to s, times ξ ln 10:

$$\frac{d\,\text{share}}{d\log_{10} N} = -\ln(1-\rho)\;\xi\ln 10\;(1-\rho)^{s(N)}$$

The first two factors are the linear reading, with −ln(1 − ρ) in place of ρ, which is the same to first order. The last factor is the share of runs not yet resonant. The rise is not a fixed amount a decade; it is that amount times what is left, and it shrinks as the share grows. A quantity that rises that way saturates — at one, when every run has a close enough neighbour — though it can take a very long time to get near.

The formula is checked against the share itself: a centred difference of the share across a one per cent change in N at a million million sites matches it to a part in a thousand at all ten concentrations.

For runs of six at x = 0.5, the concentration first studied, the rise is 1.74 points a decade over the first three decades, 1.57 over 10⁸ to 10¹², and 1.20 over 10²⁰ to 10³⁰. Over nine decades the curve looks straight, because it has bent by only a tenth; beyond them it has bent by a third. By 10⁵⁰ sites the share is 57.7 per cent, where the straight line through the first nine decades would be at 78.5 per cent and climbing to a hundred by 10⁶³.

Where half the runs are resonant #

The half-way point has an exact form too. The share is a half when (1 − ρ)^s = ½, at a separation of ln 2 / −ln(1 − ρ), and the separation reaches that value a known number of decades after its value at any reference length.

At x = 0.5 the exact share is a half at a chain of 10^40.7 sites. The straight line through 10³ and 10¹² reaches a half at 10^32.6, and the figure quoted from it before was about 10³¹. The correction is ten orders of magnitude, in the direction of making resonant pairing rarer still. At every concentration from 0.2 to 0.9 the straight line arrives early, because it keeps adding the early rise when the exact rise has already begun to fall.

Neither number describes a chain anyone could make, and the gap between them is not a rounding: it is the difference between a rise that stays at 1.64 points a decade and one that has fallen to about 1.2 by the time the share is a third. Both are statements about what the model implies — that at any physically accessible length, most runs of six are isolated levels — and the correction strengthens that conclusion rather than overturning it. What it overturns is the reading of the curve: a logarithmic rise that “does not arrive” was the start of a saturating one.

The concentration decides how soon #

The linear reading is best where ρs is smallest. The separation at a given chain length hardly depends on concentration — at a million million sites it lies between 42 and 46 for every concentration from 0.3 to 0.9 — so ρ decides it, and ρ for runs of six rises from 8 × 10⁻⁷ at x = 0.1 to a maximum of 0.011 at x = 0.75, the concentration at which a run of exactly six is most likely, and falls again beyond.

The sequence between the ends is smooth. At x = 0.4, with ρ = 1.5 × 10⁻³, the late rise is 0.87 of the early one; at x = 0.6, with ρ = 7.5 × 10⁻³, it is 0.50; at x = 0.7, 0.37. Each step up in density brings the bend earlier, because the share not yet resonant, (1 − ρ)^s, falls faster with the separation the larger ρ is, and the separation at a given length is nearly the same for all of them. A reader who measured the slope at one concentration and assumed it held at another would be wrong by a factor that depends on the second concentration, not on the first.

Where runs are rare the straight line holds: at x = 0.3, with ρ = 3.6 × 10⁻⁴, the rise over 10²⁰ to 10³⁰ is 0.97 of the rise over the first three decades. At x = 0.5 the ratio is 0.69. At x = 0.75, where runs of six are commonest, it is 0.35 — the share is rising at a third of its early rate by 10²⁰ sites — and by 10³⁰ it has passed three quarters. At x = 0.9 runs of six are less common again and the ratio recovers to 0.61.

The two sides of the peak are not mirror images. At x = 0.9 there are more low sites than at x = 0.6, but most of them sit in runs longer than six, so runs of exactly six are rarer — ρ is 5.3 × 10⁻³ against 7.5 × 10⁻³ — and the separation at a given chain length is slightly smaller because the level spacing is set by all the gap states, whose number changes too. The share at a million million sites is 20.8 per cent at x = 0.9 and 28.8 at x = 0.6. What the two concentrations share is the lesson: the approximation’s accuracy follows ρ, and ρ is not monotone in the concentration.

So the claim that the rise is ρξ ln 10 per decade is right in one regime and not a law. What controls the regime is the product of the run density and the separation at the lengths considered, which is small at low concentration at any length anyone would consider and not small near the concentration of maximum density over the same range.

The chains obey the law the curve comes from #

Everything above follows from the geometric law for the separation between runs. That law was checked in generated chains at x = 0.5; a sweep across concentration needs it checked where the runs are much rarer and much commoner.

In three chains of 200,000 sites at each of x = 0.3, 0.75 and 0.9, runs of six occur at the closed-form rate to within six per cent, and the fraction of runs whose nearest same-length neighbour lies within the separation for each threshold from two to fourteen decades below the closest coupling matches the geometric law to within four points. At x = 0.3 that rests on 212 pairs and at x = 0.75 on 6,692. The saturation is a property of the law the chains obey, not of an idealisation of it. The chains cannot test the saturation directly — at 200,000 sites the share of runs of six in resonant pairs is a few per cent at any concentration — but they test every ingredient of it: the density that sets the rate of approach, and the geometric form of the separations whose tail the saturation is. A different separation law, one with a heavier tail than geometric, would saturate later; one with a lighter tail, sooner. The chains say it is geometric, to within the scatter three chains allow, at the concentrations where the answer differs most.

How it was computed #

The share, the decay length ξ and the coupling prefactor $t_0$ are those used throughout: ξ and $t_0$ from exact diagonalisation of chains holding one pair of runs of six at separations of two to fourteen, the level spacing as the energy range of runs of three to nine divided by their number in N sites, and the share as 1 − (1 − ρ)^s at the separation where the fitted coupling equals the spacing. The concentration enters through ρ and through the number of gap states that sets the spacing; ξ and $t_0$ do not depend on it. Chain lengths run from 10³ to 10⁵⁰ at twelve values, and slopes over a range are differences of the share divided by the decades between.

The exact slope is the derivative given above, and the half-way length is the chain length at which the separation equals ln 2 / −ln(1 − ρ), reached from its value at 10¹² at ξ ln 10 per decade. The straight-line reading uses the slope between 10³ and 10¹², carried on from the share at 10¹².

The checks, run wherever these figures are drawn. At every concentration the exact slope at 10¹² matches a centred difference of the share to a part in a thousand. At x = 0.5 the late rise is under three quarters of the early rise, and at x = 0.75 under two fifths. At x = 0.5 half of the runs are resonant between 10⁴⁰ and 10⁴², where the straight line puts it between 10³⁰ and 10³³. The slowing grows with density from x = 0.3 to 0.75, and at every concentration with ρ above 10⁻⁵ the straight line reaches a half too early. In generated chains at three concentrations the run density matches ρ to six per cent and the separation law to five points. The refusal is the dilute end: at x = 0.1, with ρ below 10⁻⁶, (1 − ρ)^s is one to within two parts in ten thousand over every length used, so the exact slope must be constant to a part in a thousand there — or the curvature found elsewhere would not be the curvature of the geometric law.

Where the count stops #

The nearest neighbour only. A run counts as resonant if its nearest same-length neighbour couples strongly enough. A run with a strongly coupled neighbour on each side, or a cluster of three, is counted once, and resonance between runs of different lengths — whose levels differ — is not counted at all. The share is a count of pairs, not of the states in a band.

Lengths no chain has. Everything past about 10²⁰ sites is the model’s arithmetic. The saturation is a property of the formula the chains obey over the separations they contain, carried to separations of a hundred and more that no generated chain samples.

The exponential coupling. The coupling is fitted over separations of two to fourteen and used out to the separations the long chains imply. That extrapolation was already the weakest step, and the saturation leans on it harder, because the late decades are where the separations are largest.

And runs of six. The sweep is at one run length. The slope ratio depends on ρ, which for another length peaks at another concentration, x = L/(L + 2), and the same analysis at runs of four, the commonest, would reach its saturated regime soonest.

A line that is the start of a curve #

A curve that rises by a fixed amount per decade over nine decades is a strong invitation to call it logarithmic, and over those nine decades the call is accurate to a tenth. The trouble is what the call is then used for. A reading of a curve is a claim about the curve outside the range it was read over, and a quantity bounded by one cannot rise by a fixed amount forever. Whenever a share, a fraction or a probability looks linear in a logarithm, the right question is what bounds it, because the bound is where the line must turn.

The sweep adds the sharper version. The approximation behind the line was controlled by a small product, and a single parameter moved that product by four orders of magnitude without changing anything else. Checking a linearisation at the parameter value it was derived at says little about the regime it will be used in; moving the parameter that controls its validity says a great deal.

The same structure turns up wherever a model counts things that are rare at first and bounded at the end. A random alloy splits into two bands only if the chain is short enough, and the box a run makes is a statement about the typical run and not the longest. In each, a quantity that grows with system size was read at the sizes computed, and the reading was sound there. The questions that follow — at what size does it change character, and does it ever — are answered by the form of the bound, not by the slope of the reading.

Still open: runs of four, and the square net #

The obvious open question is the run length. Runs of four are commoner than runs of six at every concentration and have a shorter decay length, and their share was already 41 per cent at a million million sites at x = 0.5 — well into the regime where the straight line fails. Repeating the sweep for runs of four to eight would say whether the half-way length for the commonest runs falls within a range a real material’s disorder could reach, which for runs of six it never does.

The nearer question is dimension. On a square net the number of neighbours within a distance grows as its square rather than linearly, so the geometric law is replaced by one whose share rises much faster with the separation — and a share that rises faster saturates sooner. The nets for that exist already, and the question is whether the saturated regime that a chain reaches only at absurd lengths is reached on a plane at lengths that are not absurd.