Spectrum Estimation through Kirchhoff Random Forests

Simon Barthelmé; Fabienne Castell; Alexandre Gaudillière; Clothilde Mélot; Matteo Quattropani; Nicolas Tremblay

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arXiv · 2507.19164

Spectrum Estimation through Kirchhoff Random Forests

Simon Barthelmé·Fabienne Castell·Alexandre Gaudillière·Clothilde Mélot·Matteo Quattropani·Nicolas Tremblay

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Abstract

Given a non-oriented edge-weighted graph, we show how to make some estimation of the associated Laplacian eigenvalues through Monte Carlo evaluation of spectral quantities computed along Kirchhoff random rooted spanning forest trajectories. The sampling cost of this estimation is only linear in the node number, up to a logarithmic factor. By associating a double cover of such a graph with any symmetric real matrix, we can then perform spectral estimation in the same way for the latter.

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Simon Barthelmé, Fabienne Castell, Alexandre Gaudillière, Clothilde Mélot, Matteo Quattropani, Nicolas Tremblay. 2025-07-25. Spectrum Estimation through Kirchhoff Random Forests. https://arxiv.org/abs/2507.19164

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