Determinantal random subgraphs

TL;DR

The paper defines two determinantal random subgraph models: (i) connected spanning subgraphs with a fixed number of independent cycles and (ii) acyclic spanning subgraphs with a fixed number of components. Each model yields a determinantal probability measure whose kernel is a structured subspace of 1-forms, with weights expressed via determinants of small matrices involving edge-forms and cycle/coboundary data, thereby generalizing Kirchhoff and Symanzik polynomials. A unifying matroid framework shows these families are dual; the results extend to linear matroids, with partition functions given by projections in exterior powers and computable through Schur-complement type identities. The approach blends integral and geometric multilinear identities, exterior calculus on graphs, and the mean-projection properties of determinantal point processes, yielding real-stable polynomial generating functions and new perspectives on Laplacian-related forest measures. Collectively, the work provides a compact exterior-algebra formulation for a broad class of determinantal spanning-subgraph models with rich dualities and connections to physics-inspired polynomials and matroid theory.

Abstract

We define two families of determinantal random spanning subgraphs of a finite connected graph, one supported by acyclic spanning subgraphs (spanning forests) with fixed number of connected components, the other by connected spanning subgraphs with fixed number of independent cycles. Each family generalizes the uniform spanning tree and the generating functions of these probability measures generalize the classical Kirchhoff and Symanzik polynomials. We call Symanzik spanning forests the elements of the acyclic spanning subgraphs family, and single out a particular determinantal mixture of these, having as kernel a normalized Laplacian on $1$-forms, which we call the Laplacian spanning forest. Our proofs rely on a set of integral and real or complex (which we call geometric) multilinear identies involving cycles, coboundaries, and forests on graphs. We prove these identities using classical pieces of the algebraic topology of graphs and the exterior calculus applied to finite determinantal point processes, both of which we treat in a self-contained way. We emphasize the matroidal nature of our constructions, thereby showing how the above two families of random spanning subgraphs are dual to one another, as well as possible generalisations.

Figures (13)

• Figure 1: A spanning unicycle (left) and a two-component spanning forest (right) of a $15\times 15$ square grid graph.
• Figure 2: A random element of $\mathcal{C}_4(\mathsf{G})$ (its $2$-core is represented by thickened edges) and a random element of $\mathcal{F}_{4}(\mathsf{G})$ on a hexagonal grid determined by $4$ random $1$-forms (note that one of the trees has zero edges). The sampling algorithm we used here, and for the other figures, is the classical one of HKPV for sampling determinantal probability measures.
• Figure 3: Duality between the probability distributions on spanning forests and spanning connected subgraphs, illustrated here in the case $k=4$ for a square grid graph.
• Figure 4: A uniform element of $\mathcal{H}_1(X)\subset \mathcal{C}_{2}(X_{1})$ where $X=(X_0,X_1,X_2)$ is a quadrangulation of the flat torus of genus $1$.
• Figure 5: The $2$-core of a uniform element of $\mathcal{H}_1(X)\subset \mathcal{C}_{4}(X_{1})$ where $X=(X_0,X_1,X_2)$ is a discretization of a surface of genus $2$.

Theorems & Definitions (116)

• Proposition 2.1
• proof
• Proposition 3.1: First cycle-tree identity
• proof
• Proposition 3.2: Second cycle-tree identity
• proof
• Proposition 3.3: First coboundary-tree identity
• proof
• Proposition 3.4: Second coboundary-tree identity
• proof