The free uniform spanning forest
Random spanning trees of finite weighted graphs: the matrix-tree theorem, inclusion probabilities and covariances of edges through the reduced Laplacian, negative correlation, and the response identity that bounds the dependence of the inclusion probabilities on the weights uniformly in the graph. The free uniform spanning forest of an infinite connected graph with finite degrees: determinant formula for the uniform spanning tree, convergence along exhaustions, the limit law and its properties. Tilts of the forest by numbers on finitely many edges, their meaning as conditional laws, the response bound, and the drift. The integral equation of the drift for arbitrary families of continuous paths: existence, uniqueness, causality, measurability. The course is the start of an exposition of OpenAI's theorem (2026) that the free uniform spanning forest is a factor of independent labels; that theorem is not yet in the course.
Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Weighted spanning trees
- The forest as a limit of trees
- Tilted laws and the drift
- The integral equation of the drift
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Determinants and inverse matrices (core course Linear Algebra); orthogonal projections in the space of square-summable sequences (core course Functional Analysis); product measures, densities and uniqueness of a measure from its values on cylinders (core course Measure and Integration); the lessons prove the other facts they use.