The threshold a symmetry pins down

The average scatters sharply and the rest does not treats a randomly occupied structure as an average plus a departure from it, and reads the departure out of the diffuse scattering. That is a question about a mean. This essay asks a question the mean cannot answer: whether the occupied sites join up.

The question, and why it has a sharp answer #

Fill each site of a lattice independently with probability p. At small p the occupied sites form scattered islands. At large p they form one connected region spanning the crystal. Between the two there is a threshold, p_c, and the transition at it is sharp — arbitrarily close below, every cluster is finite; arbitrarily close above, an infinite one exists.

The threshold depends on the lattice, and for almost every lattice its exact value is unknown. It is estimated by simulation, quoted to a handful of digits, and no closed form for it exists. Square-lattice site percolation is the standard example: 0.5927 and a little more, with no argument that produces it.

The triangular lattice is different, and the difference is a symmetry.

Two things about that sharpness deserve stating before anything is computed, because both are easy to assume and neither is obvious. The first is that a threshold exists at all: it is a theorem, not a definition, that the probability of an infinite cluster is zero below some occupancy and one above it with nothing in between. The second is that the transition is in the occupancy and not in the size of the sample — a finite patch shows a gradual rise, and what sharpens as the patch grows is the curve rather than any single measurement. So every number below is measured on a finite box, and the argument’s job is to say what the infinite lattice does with them.

That table is not a statistical result. The claim is that on the triangular lattice the two events are complementary — for every configuration, without exception — so a single counterexample would end it. Four hundred configurations at occupancies spread from a fifth to four fifths produce none.

It is worth saying what “occupied at random, independently” excludes, because a real crystal rarely satisfies it. Independence means the occupancy of one site tells nothing about its neighbour’s, which is exactly what a solid solution with no short-range order has and exactly what the order a diffuse pattern measures measures the departure from. A structure with short-range order percolates at a different occupancy — clustering lowers the threshold and alternation raises it — and the numbers here are the reference the departures are measured against rather than predictions about any particular material.

Why the faces decide it #

The property has a name, self-matching, and a reason.

Suppose no occupied path crosses from left to right. Then something is blocking every attempt, and what blocks it must be a connected wall of vacant sites reaching from top to bottom. Whether such a wall exists depends on what “connected” means for the vacant sites, and that is where the face shape enters.

On the triangular lattice every face is a triangle, so three sites bounding a face are mutually adjacent. There is no gap for a path to slip through: an occupied path and a vacant path cannot cross without sharing a site, which is impossible since a site is one or the other. So blocking is exactly equivalent to the vacant wall existing, and the two events are complementary.

On the square lattice a face has four sites, and its two diagonals are not adjacencies. An occupied diagonal pair and a vacant diagonal pair can cross at that face with neither blocking the other. So the blocking of one crossing does not produce the other, and both can fail — which the census counts, in a quarter of its samples.

The asymmetry between the two lattices is worth stating in the direction that makes it useful. What the triangular lattice has is not an extra property that happens to be convenient; it is the absence of a way for two things to avoid each other. A square face offers a crossing without contact, and offering it once is enough to break the complementarity — the failures counted in the census are configurations in which some square face somewhere was used that way. So a self-matching lattice is one with no such face anywhere, and every triangulated lattice is self-matching for the same reason this one is.

The argument has a shape worth recognising, because this collection meets it repeatedly. A geometric question is settled by asking what a face of the structure looks like: a triangulated lattice leaves no gap for two paths to pass through, and that one local fact decides a global one. A polyhedron is two properties of a graph settles a different global question — whether a graph is a polyhedron — by two conditions of the same local kind, one of which is exactly that a certain kind of separation cannot happen. In both cases the conclusion is about the whole object and the ingredient is about a face.

From the property to the number #

The threshold follows in two sentences, and neither of them is a calculation.

At p = 1/2 occupied and vacant sites have the same distribution, so the occupied-crossing event and the vacant-crossing event are equally likely once the patch is turned through a right angle to exchange left–right with top–bottom. They are complementary, so their probabilities sum to one. Each is therefore exactly 1/2.

A crossing probability of exactly a half is the signature of being at the critical point, and it holds at every box size rather than only in the limit — which is a much stronger prediction than a threshold estimate, and a much easier one to test.

The square lattice is the control, and it behaves as a control should. Nothing about the argument applies to it, its crossing probability at a half is not pinned, and it drifts downward with size — from about a fifth at eight sites across to under one per cent at thirty-two. A quantity that drifts like that is a quantity being measured on the wrong side of a threshold.

The bisection is the honest confirmation, because it is the same code on both lattices with no knowledge of which one has a theorem. It returns a half for the triangular lattice and something near six tenths for the square one, and it cannot tell which of those is exact. Only the argument can.

How the bisection works is worth a sentence, because it is doing something a naive search would not. It does not estimate the crossing probability at many occupancies and look for where the curve passes a half; it asks one yes-or-no question at each step — is the crossing probability here above a half or below it — and halves the interval. Fourteen halvings take an interval of a half down to three parts in ten thousand, and each costs four hundred configurations rather than the far larger number a smooth curve would need. The threshold is a location and locations are found by bisection; only the curve in the next figure needs the probability itself.

The exchange step deserves one more sentence, since it is the step that uses the lattice’s symmetry rather than its faces. Turning the patch through a right angle must carry the lattice to itself for the two crossing events to be exchangeable, and on a triangular lattice drawn on a rhombus that is a symmetry of the arrangement. Without it the two events would be complementary and unequal, and their probabilities would sum to one at every p without either being a half at any particular one. So the argument needs two ingredients — a face shape and a symmetry of the patch — and the second is why the shape of the region matters and not only the shape of the lattice.

One consequence of the every-size claim is worth extracting because it is the practical test. A quantity predicted to be a half at every box size can be checked at a small box, where a simulation is cheap and its error bars are honest, and the check is as strong there as at a large one. That inverts the usual relationship between simulation and finite size, in which small boxes are the ones whose answers cannot be trusted. Here the small box is trustworthy and the large box is confirmation, and the reason is that what is being tested is an identity rather than a limit.

What this says about a crystal #

Three things, and the third is where the sharpness matters most.

The first is that a randomly substituted crystal has a connectivity threshold, and it is a property of the lattice rather than of the atoms. A solid solution in which one species conducts and the other does not becomes conducting at a composition set by the coordination of the sublattice being filled, not by the chemistry — and a structure with more neighbours per site percolates at a lower composition, which is the whole content of the two thresholds computed here. The average scatters sharply and the rest does not sees the same disorder in reciprocal space and sees nothing of the connectivity, because an average has no notion of a path.

The second is that the transition is sharp in a sense that ordering transitions are not. There is no free energy here, no temperature and no interaction: sites are occupied independently, and the sharpness is purely geometric. So percolation is the cleanest example of a threshold produced by connectivity alone, and where a real material shows a sudden onset of conduction or of rigidity at a composition, the first question is whether anything more than geometry is needed to explain it.

The third is that the same computation answers a mechanical question. A framework whose bonds are present with probability p becomes rigid at a threshold of its own, and that threshold is not the connectivity one — a network can be connected and floppy. A game that decides what counting only bounds is the machinery for deciding rigidity on a given network, and running it over random networks is how the rigidity threshold is found. The two thresholds differ, and the difference is the reason rigidity percolation is a separate subject.

The second of those is the one that keeps the argument honest. A property that held on both lattices would not be a reason for the triangular threshold to be special, so the square lattice is required to fail the complementarity test, and it does — in a quarter of its configurations, which is far too many to be an artefact of the patch size.

A fourth reading is worth adding because it connects the threshold to a quantity this collection computes elsewhere. The threshold falls as the number of neighbours per site rises — six neighbours give a half and four give six tenths — and the same monotonicity holds across dimensions and across lattices, so a rough rule is that the product of the threshold and the coordination number is nearly constant. That coordination number is the kissing number of the lattice, the count of nearest neighbours, and it is the same quantity that decides how densely spheres pack. Two questions with nothing obviously in common turn out to depend on one integer, which is worth noticing even where the dependence is a rule of thumb rather than a theorem.

One more crystallographic reading, since the sublattice matters as much as the lattice. A substituted site in a real structure is one Wyckoff orbit among several, and the connectivity that matters is connectivity within that orbit — two occupied sites are neighbours when they are near enough for whatever is being conducted, which is a chemical criterion and not a geometric one. So applying any of this to a structure means first deciding which sublattice is being filled and what counts as a contact on it, and both decisions change the coordination number and therefore the threshold. The symmetry of an average is where the difference between a site and the orbit it belongs to is set out.

Where this stops #

The argument gives one threshold and no others. Bond percolation on the square lattice is also exactly a half, by a duality argument of the same family — the square lattice is its own dual, so an open crossing in the lattice and a closed crossing in the dual are complementary in the same way. That case is not computed here, and the family resemblance is worth knowing precisely because it shows what the ingredient is: a lattice that is its own matching or its own dual, and nothing else.

There is a related result the same family of arguments gives and this essay does not, and it is worth naming so the boundary is clear. The threshold for site percolation on the hexagonal lattice is the largest of the common two-dimensional ones, and the hexagonal and triangular lattices are duals — so a duality argument relates them, but it relates bond percolation on one to bond percolation on the other, not site percolation. Site and bond percolation behave differently under duality and matching, and conflating the two is how a plausible chain of reasoning arrives at a wrong threshold. Covering and packing want different lattices is the same warning about a different pair of quantities that a lattice makes it tempting to identify.

Everything else in two dimensions is a simulation. Site percolation on the square lattice, bond percolation on the triangular and hexagonal lattices except where the star-triangle relation settles them, every lattice with more than one kind of site — all estimated, none derived. The star-triangle relation named above is the one other exact ingredient in the plane and it is worth one sentence, since it is the reason the triangular and hexagonal bond thresholds are known. It says that a triangle of bonds and a three-pointed star of bonds can be exchanged without changing any connectivity between their outer vertices, provided the probabilities are related in a particular way — and that relation, combined with duality, pins the two thresholds to expressions involving the root of a cubic. Neither is a half, both are exact, and neither is derived here.

And in three dimensions nothing at all is exact, because the duality that pairs a path with a blocking path in the plane pairs a path with a blocking surface in space, and a surface is not the same kind of object.

There is a class of questions the crossing formulation does not reach, and it is the one most often asked. What fraction of the sites belongs to the spanning cluster, how that fraction rises above the threshold, and with what exponent — these are the critical properties of the transition, they are universal in a way the threshold is not, and none of them is decided by any of the reasoning here. The self-matching argument fixes where the transition is and says nothing about how it happens. The arrangements a crystal keeps at absolute zero counts configurations rather than measuring a transition, and is the nearer neighbour of this essay in that respect: both are about how many arrangements there are, and neither is about how a system moves between them.

What is computed here is also a finite patch with hard edges, which is not the infinite lattice the threshold is defined on. The crossing-probability formulation is what makes that acceptable: it is a statement about a finite box that holds at every size, so nothing is being extrapolated. A quantity like the fraction of sites in the largest cluster would need extrapolation, and would need much more care.