The count that depends on the edge

Every count of arrangements so far has been a count per vertex. The arrangements a crystal keeps at absolute zero reports Lieb’s 1.5396 and Pauling’s 1.5 as growths per vertex; the defect census reports entropies per vertex at every defect density. That is what a residual entropy is for: a lump of ice has a number of arrangements with more digits than it has atoms, and the only usable statement is the growth per atom.

A growth per atom is supposed to forget the boundary. Doubling the crystal doubles the logarithm; the surface is a vanishing fraction and contributes nothing in the limit. That is the assumption every extensive quantity rests on, and it is stated so rarely that it is easy to take for a definition.

Square ice has two residual entropies, and the difference between them is not small.

The boundary that does it #

The second boundary condition is not a natural one for a crystal and is not offered as one. It is the condition the six-vertex model happens to be exactly solvable under, and the reason it matters here is that the answer it gives is different.

Fix every arrow on the top and bottom edges pointing inward and every arrow on the left and right pointing outward. Nothing else is fixed. Then count the arrangements of the interior arrows that satisfy the rule at every vertex.

The answers are 1, 2, 7, 42, 429, 7,436 and 218,348. Brute force reaches the third — sixteen million assignments at the fourth is more than brute force should be asked for — and the rest come from a product formula, with the two agreeing everywhere both can be asked.

Seven for a three-by-three square is small enough to see why the count is not larger. The nine vertices have twelve interior arrows between them, four thousand and ninety-six ways of drawing those, and the rule at nine vertices cuts them to seven. Most of the cutting happens at the edges, where two or three of a vertex’s four arrows are already fixed and the rule leaves one choice or none.

That sequence is one of the more famous in combinatorics and it is not from this subject. The domain-wall arrangements are the alternating sign matrices: mark each vertex whose four arrows all point in with a plus one, each whose four point out with a minus one, and the rest with nought, and the result is a square of entries whose rows and columns each sum to one with the non-zero entries alternating in sign. The arrow picture and the matrix picture are the same object, and the count is the same count.

Two limits, approached from opposite sides #

The per-vertex growths of the two sequences do not merely differ at small sizes.

The torus count per vertex is 2.060, 1.742, 1.648, 1.608, 1.586 at sides two to six, falling towards 1.5396, which is (4/3) to the power three halves. The domain-wall count per vertex is 1.189, 1.241, 1.263, 1.274, 1.281, rising towards $3\sqrt{3}/4$, which is 1.2990.

Neither sequence is converging quickly, which is the ordinary state of a count like this and is why both limits are quoted rather than extracted. At side six the torus is still three per cent above its limit and the domain wall one and a half per cent below its own.

The directions are as instructive as the numbers. A small torus counts too many, because periodic edges let an arrangement wrap round and be legal where an infinite lattice would not permit it — the same overcount a calculation in a box always carries. A small square inside a domain wall counts too few, because a small square is nearly all boundary and the boundary is where the arrangements are forced.

Both errors shrink, and they shrink towards numbers sixteen per cent apart. No larger size brings them together. The residual entropy of square ice is 0.4315 a vertex or 0.2616 a vertex, and which one depends on what was done to the edge.

Where the missing arrangements went #

The domain wall is the smaller count, and it is worth asking where the difference is rather than only that it is there.

Averaging over every one of the 7,436 arrangements at side six and asking, at each vertex, how often it carries no sign at all, gives a map with a clear shape: 0.94 at the corners against 0.61 in the middle. At side four the corner value is 0.83 and at side five 0.90, so the corners are heading for one.

The corners are not frozen at any finite size and they freeze in the limit. A region near each corner becomes, as the square grows, a region where every arrangement agrees — the arrows all pointing the same way, no choices taken. That region contributes nothing to the entropy, and it is a fixed fraction of the square rather than a vanishing one, so the per-vertex count is dragged down by a fixed factor for ever.

That is the arctic phenomenon, and it has been met here once already. A facet with no energy in it counts piles of cubes in the corner of a box, finds the average pile exactly, and observes that it has a flat corner meeting a rounded middle — a crystal’s equilibrium shape produced by counting with no surface energy anywhere in the argument. The flat corner of the pile and the frozen corner of the square are the same object. There the boundary was the walls of the box; here it is the domain wall, and in both cases the height function is pinned at the edge and has room only in the middle.

A height pinned at its edges, which is the whole mechanism #

The clearest account of why a boundary reaches into the middle is in the second description of the arrows.

An arrow field with no divergence is the rotated gradient of a height on the faces: give every face a number, and let each arrow record the difference across the edge it sits on. The rule is exactly the condition that such a height exists. So an arrangement is a surface, and counting arrangements is counting surfaces.

A boundary condition on the arrows is a boundary condition on the surface’s slope, and fixing every boundary arrow fixes the height all the way round the edge. The domain wall fixes it to the steepest profile the lattice permits — a square pyramid, rising from the corners — and a surface pinned to that has almost no room near the corners and a good deal in the middle. The torus pins nothing and the surface is free to be rough everywhere, which is the roughness the pinch point measures.

That is also why the missing entropy is a fixed fraction rather than a surface effect. A pinned corner does not constrain a strip of faces near the edge; it constrains a wedge, because the steepest profile leaves no slack anywhere inside the region where the height is already at its extreme. The wedge is a fraction of the square and stays one however large the square is.

The pile of cubes is the same statement with the surface drawn rather than inferred. There the height is the number of cubes at each place, the box’s walls pin it, and the average pile comes out flat at the corner and rounded in the middle — a shape produced by counting, with no surface energy anywhere in the argument. Here the height is not drawn and the shape appears only as a map of how often each vertex carries a sign, but the object is the same and so is the reason for it.

Every other count of this kind has a boundary in it too #

Once the question is asked it has to be asked of everything else, and the answers are not all the same.

The dimer count that opens the subject is on a rectangle with free edges, and its growth per site is Catalan’s constant over π in the limit, which is the accepted bulk value. A dimer covering has no conserved flow — the rule is that every site is used once, not that something is balanced across a cut — so a boundary condition there really does cost only a boundary’s worth.

The lozenge tilings of the pile of cubes are the opposite case and they were already known to be: the hexagon’s boundary is what makes the arctic curve, and the count inside a hexagon is famously not the count on a torus. That essay’s whole point is the shape the boundary produces, so its boundary-dependence is the result rather than a caveat.

Square ice was the case where it was not obvious, because the rule is local, the lattice is the plainest one there is, and the number 1.5396 is quoted as though it were a property of the substance. It is a property of the substance and a torus.

The census of what a defect costs is on a torus throughout, so every entropy in it carries the same qualification — and the dip it measures, which is a statement about the shape of a curve rather than about its height, is the kind of statement that survives a change of boundary. Which of that page’s numbers would move and which would not is a question it does not ask and this one raises.

What extensive means, and where it fails #

The usual argument for a quantity being extensive is not wrong; it is being applied where its hypothesis fails.

Take two lattices and join them. If the arrangements of the whole are the arrangements of the parts multiplied, then the logarithm adds and the growth per vertex is common to both. That is the argument, and its hypothesis is that joining costs only a surface’s worth of arrangements — a term growing as the boundary rather than as the volume.

With a local rule and a conserved quantity, joining can cost a volume’s worth. The ice rule makes the arrow field divergence-free, so the flow across any cut is determined by what is inside it, and a boundary condition that fixes the flow at the edge fixes a quantity that reaches all the way in. The domain wall fixes the maximum possible flow through the square — every arrow in at the top, every arrow out at the sides — and the only arrangements consistent with it are the ones in which the flow is carried straight through, which is what freezes the corners.

So the failure is not general. It is a failure for rules with a conservation law in them, which is the class the ice rule belongs to, and the shape of it can be stated in one sentence: a conserved quantity turns a boundary condition into a bulk constraint. The correlations fall as a power for the same reason, and the pinch point in the scattering is the same fact in a third disguise.

Where the exactness stops #

Computed here. The torus counts by the transfer matrix the ice count already uses, to side six. The domain-wall counts by the product formula to side seven, and by enumerating every assignment of the interior arrows to side three, with the two compared. Every alternating sign matrix at sides up to six, enumerated row by row and counted, agreeing with the formula at every size. And the average over all of them of how often each vertex carries no sign.

The product formula is quoted, not derived. It gives the numbers 1, 2, 7, 42 and so on in closed form and its proof is one of the harder results in enumerative combinatorics. What is checked here is that it reproduces the brute-force count where brute force can run, and that its own enumeration by rows gives the same totals.

The two limits are quoted as well. Lieb’s (4/3) to the three halves and the domain wall’s $3\sqrt{3}/4$ are exact results for infinite lattices and nothing on this page proves either. The sequences computed here are consistent with both and are far too short to establish either on their own.

The torus’s own overcount is not a small correction. At side six the per-vertex count is 1.586 against a limit of 1.5396, which is three per cent — larger than the two and a half per cent by which Pauling’s estimate misses the same limit. A reader comparing an approximation against a finite exact count is therefore comparing two errors of similar size pointing opposite ways, and the sizes reachable here are all in that range. The limits are quoted for exactly that reason.

And nothing here is a crystal. A domain wall is a boundary condition chosen for solvability, not one any experiment imposes. What is real is the moral, which is about what a residual entropy is: a quantity a calorimeter measures on a lump with a surface, and the theory that predicts it makes an assumption about that surface which it usually does not state.

The refusal is the whole essay in one line: a residual entropy per vertex offered without saying which boundary it was counted on. Everything else on this page is evidence that the omission changes the answer.

Who counted, and what it was counted for #

William Mills, David Robbins and Howard Rumsey conjectured the product formula for the alternating sign matrices in 1983, while counting something else entirely — a determinant evaluation arising from Charles Dodgson’s condensation method, which is to say from Lewis Carroll’s. Doron Zeilberger proved it in 1996 in a paper of eighty-four pages, and Greg Kuperberg gave a much shorter proof the same year by recognising the alternating sign matrices as the six-vertex model with domain-wall boundaries and using Anatoli Izergin’s determinant formula for its partition function.

Kuperberg’s step is the one this page rests on, and it is worth stating for what it says about the subject: a combinatorial identity that had resisted proof for thirteen years fell to a change of picture, from matrices to arrows, and the arrows were a physics model from 1967 that nobody had connected to it. The arctic curve for domain-wall ice was established later still, by Colomo and Pronko and others.

The entropy comparison on this page is not anybody’s result. It is two known numbers set beside each other, and the reason it is worth setting them beside each other is that they are usually quoted in different subjects.

Still open: how much boundary is enough #

The two boundaries here are the two that are exactly solvable, which is a small sample of a large question.

Which boundary conditions freeze corners and which do not? The torus has no boundary and freezes nothing. The domain wall is the extreme case, fixing the maximum flow. Between them lies every mixture — a square with domain walls on two sides and free arrows on the other two, a torus with one direction cut — and the growth per vertex presumably runs between the two limits. Whether it varies continuously with the fraction of the boundary that is fixed, or jumps, is not something this page can see: the transfer matrix that gives the torus count and the enumeration that gives the domain-wall count are different machines, and a mixed condition needs a third.

And how much of the lattice is frozen? The corner regions are a fixed fraction of the square, and the fraction is what the difference in entropy is made of. A free-energy argument would recover it from the two limits; a direct measurement would extend the map of how often each vertex carries a sign to sizes where the frozen region is unmistakable rather than suggested. At side six the corner is at 0.94 and the middle at 0.61, which is a gradient and not yet a curve.