Discrete Extremal Length

We consider a discrete analogue of the well-known conformal invariant – the extremal length of a family of curves on a plane.
The discrete extremal length of a family of paths $\Lambda$ on a square lattice is
$$EL(\Lambda)=\sup\limits_g \inf\limits_{P\in\Lambda} \frac{(\sum\limits_P g_i)^2}{\sum g_i^2}.$$

Here the sum in the denominator is taken over all the edges of the lattice; $\sup$ is taken over all nonnegative functions $g$ defined on the edges of the lattice, such that the denominator isn't equal to zero or infinity.
In this talk we will consider a view on the discrete extremal length related to uniform spanning trees (UST). It's the most simple and elegant way of calculating the extremal length and constructing the discrete analytical functions.
In fact, discrete extremal length helps to construct conformal mappings and it can be used to prove uniformization theorems. Generally a uniformization theorem states that an arbitrary domain of a plain can be mapped onto some domain of canonical form. The most well-known example of such a theorem is the Riemann mapping theorem.
We will discuss a uniformization of quadrilaterals onto rectangles and make several remarks concerning multiple connected domains.