Optimality Notions for Resolvent Monte Carlo

Resolvent Monte Carlo estimates eigenvalues of large matrices by sampling Markov chains and reading the target value off a truncated resolvent quotient, trading exact arithmetic for a stochastic error that the almost-optimal sampling scheme is designed to suppress. We study when that error vanishes outright. We derive an exact closed-form identity for the variance of the moment estimators of a general, possibly signed matrix, and use it to isolate a hierarchy of zero-variance notions ranging from the most local, which constrains only the first draws, through the finite-truncation regime that a practical run can certify, to the global regime in which every moment estimator is deterministic. We separate determinism of the estimator from correctness of the eigenvalue it reports, and exhibit the exact conditions under which each notion holds and the examples that separate them. A single edgewise condition, which we call the eigen-triple condition, forces the truncated quotient to equal the target eigenvalue in finite samples; the associated moment and quotient variances are second order in the maximal edge defect and vanish at the eigen-triple. A linear-time procedure certifies the condition.