Lieb's square ice constant

Binary 1.10001010001000110100010111001100…
Decimal 1.53960071783900203869106341467188…
Hexadecimal 1.8A2345CC04425BC2CBF57DB94EDCA6B2…
Continued fraction 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{5 + \cfrac{1}{1 + \cfrac{1}{4 + \ddots}}}}}
Algebraic form \frac{8\sqrt{3}}{9}

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

\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.

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

See also

References

  1. ↑ Lieb, Elliott (1967). "Residual Entropy of Square Ice". Physical Review 162 (1): 162. doi:10.1103/PhysRev.162.162.
  2. ↑ (sequence A118273 in OEIS)