Math 2.0

Statistical mechanics, structure of matter Exactly solved models Theorem

Lieb's square ice constant

Builds on Six-vertex model

Statement

For N≥1N \ge 1, consider the N×NN \times N square grid on a torus. Let f(N)f(N) be the number of orientations of its edges such that every vertex has exactly two incoming and two outgoing edges. Then lim⁡N→∞f(N)1/N2=(43)3/2=839=1.5396…\lim_{N \to \infty} f(N)^{1/N^2} = \left( \frac{4}{3} \right)^{3/2} = \frac{8 \sqrt{3}}{9} = 1.5396\ldots

Remarks

The orientations counted are the configurations of the six-vertex model with all weights equal to 11. Lieb proved the formula in 1967 by writing f(N)f(N) through a transfer matrix and finding its largest eigenvalue with the Bethe ansatz. Pauling's 1935 estimate gives 4⋅616=324 \cdot \frac{6}{16} = \frac{3}{2}: each vertex allows 66 of the 1616 orientations of its four edges, and there are two edges per vertex. The exact constant is a little larger than this estimate. The same constant governs the growth of the number of proper 33-colourings of the faces of the square grid, since such colourings correspond to ice configurations.

Going deeper

For a student

This was the first exact solution of a six-vertex model, and it opened the way to the general solution of ice-type models. The method, a transfer matrix diagonalized by the Bethe ansatz, became a standard tool for exactly solvable models. The correspondence with 33-colourings makes the result also a theorem of enumerative combinatorics.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Graph, concept, no evidence levelGraphconceptTransfer matrix method, concept, no evidence levelTransfer matrix methodconceptSix-vertex model, example, no evidence levelSix-vertex modelexampleLieb's square ice constant, statement, no evidence levelLieb's square ice con…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Graph concept
  2. Transfer matrix method concept
  3. Six-vertex model example
  4. Lieb's square ice constant statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Lieb%27s_square_ice_constant
About this page
TagNXA6N97CXN
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Kindstatement, theorem
WritingW0 stub
MSC82B23, 05A16, 82B20
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AboutSix-vertex model
UsesSix-vertex model
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Changed3 November 2026

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