Lieb's solution concerns the entropy of square ice with NO boundary conditions, or equivalently, with maximum entropy boundary conditions; I believe that maximum entropy boundary conditions in this case means that the arrows along each side of the square alternate in-out-in-out-... (This may even be a theorem: Scott Sheffield would know.) ASM's (alternating-sign matrices), as studied by Mills, Robbins, Rumsey, Zeilberger, Kuperberg, and many others, correspond to a very special subclass of states of the square ice model: those with "domain-wall boundary conditions", with all arrows pointing inward along the left and right sides of the square and outward along the top and bottom sides of the square. This subclass does not have full entropy; its growth rate is governed by a strictly smaller constant. Jim Propp On Tuesday, February 25, 2014, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Mmm... I have essayed a cursory inspection of Lieb (1967).
The first caveat is that it is upholstered with technical jargon that only another math. phys. specialist (which I surely have no claim to be) might be expected to follow. Nonetheless, I have come away with a strong impression of what has and has not been achieved. Bear in mind that a physicist's or engineer's concept of what constitutes "proof" is in general rather different from what a mathematician understands by the term --- not unreasonably, given that any intended application necessarily relies on intuitive judgement based on approximate experiment.
Another difficulty is that the notation actually varies between sections: eg. "N" in section I corresponds to "N^2" in section II . More seriously, what is intended by the term "subspace" appears to have little connection with its customary meaning. And Jim Propp has already drawn attention to a lack of clarity concerning the invocation of the "Bethe ansatz". I want to try to establish the importance of such complaints.
The proof depends on investigating the `transfer' matrix, that is the adjacency matrix of a graph, having 2^m vertices representing columns in an m x n Hardin array (grid 3-colouring, etc. etc.), with edges between columns which can validly lie adjacent. Under a suitable change of basis this matrix decomposes over the rationals into a number of smaller blocks: the corresponding subtotals, into which T(m, n) is partitioned by final column, form sequences as n varies which lie in a subspace associated with (remarkably) just one of these blocks.
The first stage is to show that the block concerned is (some) one on which T(m, n) depends -- or to put it another way, that in the explicit formula as a sum of exponentials in n , the maximal eigenvalue gets a nonzero coefficient. Despite the plethora of wave-mechanical legerdemain that occupies sections II and III, I can find no evidence that the author has actually nailed this; indeed, at the start of section IV he seems to come close to admitting that it remains unsubstantiated.
The second stage is to determine the maximum eigenvalue of the matrix, to which section IV devotes an impressive analytical manipulation which can quite possibly be made transparent, and in any case fairly certainly has managed to arrive at the correct answer: c = (4/3)^(3/2) .
But since (presumably) we now have to hand the Mills-Zeilberger formula with asymptotic behaviour equivalent to T(n, n) ,
f(n) = 1! 4! ... (3n-2)! / n! (n+1)! ... (2n-1)!
surely somebody whose calculus skills are less atrophied than my own can just roll up sleeves, take the logarithm, approximate the sums by integrals and deduce that f(n) ~ c^(n^2) ?
Fred Lunnon
On 2/24/14, Cris Moore <moore@santafe.edu <javascript:;>> wrote:
The simplicity of Lieb's constant (4/3)^(3/2) has long made me dream of
an
elementary combinatorial proof, say at the same level of technicality as the entropy of domino tilings of the square lattce (Jim has a lovely review paper about this, which I cribbed from for my book).
But the 6-vertex ice model and the Bethe ansatz seems totally different from the permanent-determinant trick we can use for planar perfect matchings (dominos, rhombi, etc). To put it differently, there seem to be at least two reasons why a stat mech model can be "exactly solvable", and they seem incomparable to each other.
Cris
On Feb 24, 2014, at 1:25 PM, James Propp <jamespropp@gmail.com<javascript:;>> wrote:
I should say that I've never read Lieb's paper. I did once try to read Rodney Baxter's proof of Lieb's result, but there was one place (an appeal to a broad principle called the "Bethe ansatz" that in some contexts is a theorem and in others is merely a universally-believed conjecture) where I couldn't supply the missing details.
That was twenty years ago.
At some point I intend to try to find out (probably via MathOverflow) whether anyone's created a write-up that fills in all the details. My gloomy guess is that the answer is "no": all the people who'd be qualified to write such an article have other projects that they're more excited about.
I'd be happy to be wrong about this!
Jim Propp
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