Lieb's square ice constant

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

is Lieb's square ice constant.

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