Lieb's square ice constant
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.[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
- [math]\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 }[/math][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
References
- ↑ Lieb, Elliott (1967). "Residual Entropy of Square Ice". Physical Review 162 (1): 162. doi:10.1103/PhysRev.162.162. Bibcode: 1967PhRv..162..162L.
- ↑ (sequence A118273 in the OEIS)
- ↑ Ballinger, Brad; Damian, Mirela (2015), "Minimum forcing sets for Miura folding patterns", Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, pp. 136–147, doi:10.1137/1.9781611973730.11
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