Lieb's square ice constant
| Binary | 1.10001010001000110100010111001100… |
| Decimal | 1.53960071783900203869106341467188… |
| Hexadecimal | 1.8A2345CC04425BC2CBF57DB94EDCA6B2… |
| Continued fraction | |
| Algebraic form |
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
![{\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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/146be733277faf0518d5db191a4a010dee25e4d9)
is Lieb's square ice constant.
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.
- ^ (sequence A118273 in the OEIS)
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