The uniform spanning tree on Z2 is just one point of a two-parameter family of measures on "essential spanning forests" in Z2. These are spanning forests with a positive density of component trees, each with two ends. These measures first appeared (in disguised form) in work of Carroll/Speyer on the cube recurrence, and work of Fomin on the loop-erased walk.

We explain how these measures arise naturally in the study of the graph Laplacian (for planar graphs). We then construct scaling limits of spanning forest processes on multiply connected domains with fixed boundary connections and paths having prescribed homotopy type. The resulting limit objects are singular measured foliations. Speaker(s): Richard Kenyon (Brown University)