Discrete extremal length on graphs and square tilings

The notions of extremal metric and extremal length can be develop on the discrete context of finite graphs. In particular, following an article by Oded Schramm (1993), they are the main tool to build a correspondence between the 1-skeleton of triangulations of a quadrilateral and square tilings: the squares are associated to the vertices in a combinatorial fashion to fill a rectangle with no overlaps. The extremal metric expresses the length of the edge of the squares. Furthermore, extremal length can be considered on trees. It is in some sense the reciprocal of the notion of capacity from potential theory.