Lieb's square ice constant

Template:Short description Template:Infobox non-integer number Lieb's square ice constant is a mathematical constant used in the field of combinatorics to quantify the number of Eulerian orientations of grid graphs. It was introduced by Elliott H. Lieb in 1967.

Definition

An n × n grid graph (with periodic boundary conditions and n ≥ 2) has n² vertices and 2n² edges; it is 4-regular, meaning that each vertex has exactly four neighbors. An orientation of this graph is an assignment of a direction to each edge; it is an Eulerian orientation if it gives each vertex exactly two incoming edges and exactly two outgoing edges.

Denote the number of Eulerian orientations of this graph by f(n). Then

$\underset{n\to \mathrm{\infty }}{\lim }\sqrt[n^2]{f(n)}={\left(\frac{4}{3}\right)}^{\frac{3}{2}}=\frac{8\sqrt{3}}{9}=1.5396007\dots$

is Lieb's square ice constant. Lieb used a transfer-matrix method to compute this exactly.

The function f(n) also counts the number of 3-colorings of grid graphs, the number of nowhere-zero 3-flows in 4-regular graphs, and the number of local flat foldings of the Miura fold. Some historical and physical background can be found in the article Ice-type model.

See also

• Spin ice
• Ice-type model

References

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