{
  "id": "the-free-uniform-spanning-forest",
  "title": "The free uniform spanning forest",
  "description": "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.",
  "units": [
    {
      "id": "weighted-spanning-trees",
      "title": "Weighted spanning trees",
      "reader": "weighted-spanning-trees.html",
      "source": "src/weighted-spanning-trees.md",
      "source_sha256": "52f5c877486b204017d2efc7262c1a99c0f29e8c7b9167a163f7336ceae380b3",
      "status": "author_self_checked"
    },
    {
      "id": "the-forest-as-a-limit-of-trees",
      "title": "The forest as a limit of trees",
      "reader": "the-forest-as-a-limit-of-trees.html",
      "source": "src/the-forest-as-a-limit-of-trees.md",
      "source_sha256": "4d39193dbb0d2ec9e7764b0486facf4350d47f7c37015b0807de5e7554f54bd9",
      "status": "author_self_checked"
    },
    {
      "id": "tilted-laws-and-the-drift",
      "title": "Tilted laws and the drift",
      "reader": "tilted-laws-and-the-drift.html",
      "source": "src/tilted-laws-and-the-drift.md",
      "source_sha256": "faea0731d7c89db32617cdd8d9a63a656fcc88e75f63b67e5cba46f5962ced06",
      "status": "author_self_checked"
    },
    {
      "id": "the-integral-equation-of-the-drift",
      "title": "The integral equation of the drift",
      "reader": "the-integral-equation-of-the-drift.html",
      "source": "src/the-integral-equation-of-the-drift.md",
      "source_sha256": "463908ee03145a81ed8546022abe429379fc87c81a38ea546f9871495e72c7e3",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Linear Algebra (B40): determinants, expansion along a row, inverse matrices, symmetric matrices",
    "core course Functional Analysis (D20): the Hilbert space of square-summable sequences, orthogonal projections onto closed subspaces",
    "core course Measure and Integration (D10): Lebesgue measure and integral, monotone convergence, image measures, product measures and densities, measurable functions and their upper limits, the monotone class theorem and the uniqueness of a measure from its values on a generating family closed under intersections",
    "the lessons prove every other fact they use"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}