I will discuss a representation for the magnetization field of the critical two-dimensional Ising model in the scaling limit as a random filed using an ensemble of measures on the plane associated with renormalized cluster areas.

The renormalized areas come from the scaling limit of critical FK (Fortuin-Kasteleyn) clusters and the random field is a convergent sum of the area measures with random signs. The representation is based on the interpretation of the lattice magnetization as the sum of the signed areas of clusters. If time permits, potential extensions, including to three dimensions, will also be discussed. The talk will be based on joint work with F. Camia (PNAS 106 (2009) 5457-5463) and on work in progress with F. Camia and C. Garban.