Lieb's square ice constant

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.^[1]

Lieb's square ice constant

Representations
Decimal	1.53960071783900203869106341467188…
Algebraic form	${\displaystyle {\frac {8{\sqrt {3}}}{9}}}$

A Eulerian orientation of a 4×4 periodic grid graph. The orientation (assignment of a direction to the edges of an undirected graph) is Eulerian because every vertex has the same number of edges going to it as leaving it (their indegree and outdegree is equal).

For the number of possible Eulerian orientations of an n × n periodic grid graph denoted ${\displaystyle f(n)}$, Lieb's square ice constant is the limit of ${\displaystyle {\sqrt[{n^{2}}]{f(n)}}}$ as ${\displaystyle n}$ approaches infinity.

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

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

See also

• Spin ice
• Ice-type model