Kirchhoff Random Rooted Forest Trajectories

• Kirchhoff random rooted spanning forest trajectories are defined by sampling dynamic forests on graphs using determinantal formulas derived from Laplacian spectra.
• They employ methods like Wilson’s algorithm and stochastic dynamics to model forest growth, fragmentation, coalescence, and spectral estimation.
• Applications span network connectivity analysis, hitting time estimation, and efficient spectral moment inversion in large-scale graphs.

A Kirchhoff Random Rooted Spanning Forest Trajectory refers to the evolution or sampling of random rooted spanning forests in a finite or infinite graph according to probability laws derived from determinantal formulas involving the graph Laplacian and its spectrum. This field unites combinatorial graph theory, algebraic spectral theory, probabilistic methods, and random walks. Trajectories arise in two main settings: (1) the Markov or algorithmic progression of the forest state under stochastic dynamics (e.g., repeated sampling, forest growth, forest fragmentation-coalescence), and (2) the sequential mapping of nodes to roots under coupled random forest samples—each analysis informs aspects of network connectivity, spectral estimation, probabilistic hitting times, and structural properties.

1. Determinantal Measures on Rooted Spanning Forests

The fundamental object is the random rooted spanning forest, a subgraph including all vertices such that each connected component is a tree with a marked root. The Chebotarev–Shamis Forest Theorem states that for a graph with Laplacian $L$, the number of rooted spanning forests equals $\det(1+L)$ (Knill, 2013). More generally, this extends to edge-$k$-colored forests: $\det(1+kL)$ counts forests with branches colored by $k$ types. These formulas arise by expanding determinants in the basis of minors of the graph’s incidence matrix, providing a deep algebraic–combinatorial correspondence.

The probability measure on forests can be enriched: given a parameter $q>0$ and edge weights, the measure on forests $\varphi$ is

$$P_q(\varphi) = \frac{q^{|\rho(\varphi)|}w(\varphi)}{Z(q)},$$

where $|\rho(\varphi)|$ is the number of roots, $w(\varphi)$ the product of edge weights, and $Z(q) = \det(qI - L)$ (Avena et al., 2013, Avena et al., 2017). This determinantal partition function shows that combinatorial observables for forests (such as total root count) are spectral averages of the Laplacian eigenvalues.

2. Trajectories: Sampling, Dynamics, and Spectral Estimation

a) Sampling via Trajectory Composition:

Wilson's algorithm (and its variants) enables exact sampling of the forest measure for arbitrary $q$. For each node, one determines its “root” by executing a killed random walk, possibly incorporating loop-erasure; repeating this with independently sampled forests, the compositions of root maps generate “trajectories” of nodes under successive forests. For example, let $r_{\varphi_q}$ denote the root map from a sampled forest at parameter $q$; composing $l$ such maps yields the trajectory

$$\rho_l = r_{\varphi_q^{(l)}} \circ\cdots\circ r_{\varphi_q^{(1)}}.$$

The set of fixed points of $\rho_l$ estimates functionals of the spectrum:

$$\mathbb{E}\left[\frac{1}{n}\left|\{i:\rho_l(i)=i\}\right|\right] = \frac{1}{n}\sum_{j=1}^n \left(\frac{q}{q+\lambda_j}\right)^l,$$

with $\{\lambda_j\}$ spanning the Laplacian spectrum (Barthelmé et al., 31 Mar 2025, Barthelmé et al., 25 Jul 2025). In this sense, trajectories encode the rational moments of the spectral distribution, allowing for efficient Monte Carlo estimation of global spectral quantities.

b) Markovian and Fragmentation–Coalescence Dynamics:

In continuous-time models, forest dynamics are defined in which edges are added (merging trees) or removed (fragmenting trees) under prescribed rates preserving the $q$-weighted forest measure’s stationarity (Avena et al., 2013). As $q$ varies, the forest ensemble evolves through a fragmentation–coalescence process, with the number of trees (roots) increasing as $q$ grows and decreasing as $q$ decays. These processes supply an ergodic and dynamic viewpoint that links trajectories to mixing and hitting properties of the network.

3. Spectral and Probabilistic Connections

Trajectories in random Kirchhoff forests provide unbiased estimators for nonlinear spectral moments:

$$h(q, k) = \frac{1}{n}\mathrm{Tr}\left(K_q^k\right),\quad K_q = q(qI+L)^{-1},$$

and moments of the transformed empirical measure $$\nu_q = \frac{1}{n}\sum_{i=1}^n \delta_{q/(q+\lambda_i)}$$ (Barthelmé et al., 31 Mar 2025, Barthelmé et al., 25 Jul 2025). The collection of all such moments suffices to reconstruct the spectral density via inverse moment techniques (e.g., maximum-entropy methods). The approach is computationally efficient; the number of sampled forests needed scales sublinearly with the number of edges.

For hitting times, determinantal formulas relate the expected hitting time of root sets in forests to the Laplacian spectrum:

$$\mathbb{E}_x[T_R] = \frac{1}{q}\left(1 - P(|\rho(\Phi_q)|=1)\right),$$

for any starting vertex $x$ when $R$ is sampled as the root set of a random forest (Avena et al., 2013). Combinatorial identities for the discrete Green’s function and hitting times are also derived via forest enumeration (Chung et al., 2021).

4. Algorithmic Methods: Wilson’s Algorithm and Cycle-Popping

Sampling random rooted spanning forests leverages generalizations of Wilson’s algorithm. For $q>0$, the process uses loop-erased random walks that are killed at an independent exponential time. Cycle-rooted spanning forest (CRSF) measures, associated to Laplacian determinants with unitary connections, can be sampled using “cycle-popping” algorithms which stochastically decide whether to erase or keep a cycle encountered by the random walk (Kassel et al., 2012, Constantin, 2023).

For the estimation of spectral features, repeated sampling and “coupled forest” trajectories provide sufficiently rich observables to perform moment inversion even for large graphs (Barthelmé et al., 31 Mar 2025, Barthelmé et al., 25 Jul 2025). For general symmetric matrices (not strictly Laplacians), a double cover construction enables similar estimation by embedding the matrix as a signed Laplacian on an extended graph (Barthelmé et al., 25 Jul 2025).

5. Scaling Limits and Connections to Continuum

In two-dimensional or surface-embedded graphs, the scaling limit of CRSF measures leads to probability distributions on multicurves, independent of the discrete approximation sequence (Kassel et al., 2012). The scaling limit of key processes (such as loop-erased random walk) is identified with $\mathrm{SLE}_2$, as established in a series of works on conformal invariance (Kassel et al., 2012). Tightness of the sequence of measures is proved using uniform Hölder regularity of random curves.

Further, under Barycentric refinement (graph subdivision), normalized indices $\frac{\log\,\det(1+K)}{n}$ (forest index), $\frac{\log\,\det(K)}{n}$ (tree index), and their difference (tree–forest index) converge to universal quantities depending only on the maximal clique dimension, with explicit evaluation in one dimension as $2\log\phi$ where $\phi$ is the golden ratio (Knill, 2022).

6. Generalizations: Rotor Walks, Moran Model, and Nonequilibrium Applications

Rotor walks—deterministic analogues of random walks whose limiting configurations also sample the wired uniform spanning forest (WUSF) measure—embody deterministic realizations of random spanning forest trajectories (Chan, 2018, Chan, 2019). The Moran forest model, a Markov chain where vertices are repeatedly disconnected and reattached, has a stationary distribution on rooted forests with precisely characterized degree and tree-size statistics; the largest tree size scales as $\alpha\log n$ with $\alpha\approx 2.18$ (Bienvenu et al., 2019). In nonequilibrium statistical mechanics, forest representations facilitate the analysis of stationary measures, currents, and quasi-potentials in Markov jump processes (Khodabandehlou et al., 2022).

Determinantal point process (DPP) theory underpins much of the formulation: the set of forest roots in the $q$-random forest ensemble forms a determinantal process with kernel directly related to killed random walk transition probabilities (Avena et al., 2013, Avena et al., 2017). This connection allows for efficient sampling and analytic computation of multi-point statistics and correlation decay.

7. Infinite Volume and Phase Transitions

On infinite graphs, the limiting Gibbs measures for CRSFs can be constructed as weak limits of finite-volume measures with prescribed boundary conditions. Key properties include that all components with a cycle are almost surely finite, and long-range correlations decay exponentially (Constantin, 2023). In contrast, WUSF models exhibit components of potentially infinite size, and their end-structure matches that of the underlying graph in the unimodular case (Engelenburg et al., 2023). Sampling algorithms and probabilistic representations (e.g., through interlacement processes or forest trajectories) govern these limits and phase behaviors, suggesting underlying universality in the trajectory structure between discrete and continuum stochastic graph models.

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These results collectively position Kirchhoff random rooted spanning forest trajectories at the intersection of spectral graph theory, stochastic process analysis, and combinatorial probability, with diverse applications in network design, statistical physics, and efficient large-scale spectral estimation.