Lieb's square ice constant

Lieb's square ice constant

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

Lieb's square ice constant is a mathematical constant used in the field of combinatorics to approximately count Eulerian orientations of grid graphs. It was introduced by Elliott H. Lieb in 1967. It is called the square ice constant because the orientations that it counts arise in statistical mechanics of crystalline structures as the states of an ice-type model on a square grid.

The value of Lieb's square ice constant is $${\displaystyle {\frac {8{\sqrt {3}}}{9}}\approx 1.5396.}$$ Based on this, the number of Eulerian orientations of an ${\displaystyle n\times n}$ grid is $${\displaystyle \left({\frac {8{\sqrt {3}}}{9}}-o(1)\right)^{n^{2}},}$$ where the ${\displaystyle o(1)}$ term, an instance of little o notation, hides parts of the formula that tend to zero in the limit as ${\displaystyle n}$ grows.