Punctured Wilson--Evolving Sets and Root Identities for Massive Kirchhoff Forests

We construct a punctured Wilson--evolving-set coupling for finite reversible Markov chains.

The transformed evolving set carries a Markov trajectory up to its exit from a punctured domain, whereas the ordinary evolving set gives the corresponding probabilities after de-biasing.

Applied to Wilson's algorithm with exponential killing, the final-point projection yields the root law and an ordered factorization of the probability that several prescribed vertices are roots.

We identify this factorization with a probabilistic Schur-complement decomposition of the known determinantal root formula.

The survival projection yields an evolving-set representation of hitting probabilities before killing.

This representation gives a quantitative consequence which does not follow from the root process alone.

On graphs of polynomial growth satisfying a Gaussian heat-kernel upper bound, in dimension larger than four, we obtain exponential localization at scale $q^{-1/2}$ for two-point forest connectivity, up to the natural finite-volume correction, and the bound $\mathbb E[|C_q(x)|]\leq Cq^{-2}$.

Dirichlet eigenvalues in successively punctured domains also give product bounds for prescribed root events.

We record a determinant-free Poisson-type concentration bound for the number of roots, while making explicit that the determinantal description gives the sharper Bernoulli decomposition.

The complete graph is computed exactly and discrete tori, bottleneck graphs and the hypercube are treated as examples.