Lieb's square ice constant

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

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

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.[3] Some historical and physical background can be found in the article Ice-type model.

• Spin ice
• Ice-type model