Lieb's square ice constant is a

mathematical constant A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...

used in the field of

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...

to quantify the number of

Eulerian orientation In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends ...

s of

grid graph In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space , forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a latti ...

s. It was introduced by

Elliott H. Lieb Elliott Hershel Lieb (born July 31, 1932) is an American mathematical physics#Mathematically rigorous physics, mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, Cond ...

in 1967.

Definition

An ''n'' × ''n'' grid graph (with

periodic boundary conditions Periodic boundary conditions (PBCs) are a set of boundary conditions which are often chosen for approximating a large (infinite) system by using a small part called a ''unit cell''. PBCs are often used in computer simulations and mathematical mode ...

and ''n'' ≥ 2) has ''n''² vertices and 2''n''² edges; it is 4-regular, meaning that each vertex has exactly four neighbors. An

orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building de ...

of this graph is an assignment of a direction to each edge; it is an

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 :

\left(\frac\right)^\frac=\frac=1.5396007\dots

is Lieb's square ice constant. Lieb used a

transfer-matrix method In statistical mechanics, the transfer-matrix method is a mathematical technique which is used to write the partition function into a simpler form. It was introduced in 1941 by Hans Kramers and Gregory Wannier. In many one dimensional lattice m ...

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 The is a method of folding a flat surface such as a sheet of paper into a smaller area. The fold is named for its inventor, Japanese astrophysicist Kōryō Miura. The crease patterns of the Miura fold form a tessellation of the surface by par ...

. Some historical and physical background can be found in the article

Ice-type model In statistical mechanics, the ice-type models or six-vertex models are a family of vertex models for crystal lattices with hydrogen bonds. The first such model was introduced by Linus Pauling in 1935 to account for the residual entropy of water ice. ...

References

Mathematical constants Quadratic irrational numbers {{combin-stub

Definition

See also

References