Lieb's square ice constant explained

Lieb's square ice constant
Decimal:1.53960071783900203869106341467188…
Algebraic:
8\sqrt{3
}

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]

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

\limn

2]{f(n)}=\left(4
3
\sqrt[n
3
2
\right)=
8\sqrt{3
}=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

Notes and References

  1. Lieb. Elliott. Residual Entropy of Square Ice. Physical Review. 162. 1. 162. 1967. 10.1103/PhysRev.162.162. 1967PhRv..162..162L .