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arXiv · 2511.14026

Extremes of the zero-average Gaussian Free Field on random regular graphs

Lisa Hartung·Andreas Klippel·Christian Mönch
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Abstract

We study the extreme value statistics of the zero-average Gaussian free field (GFF) on random $r$-regular graphs and the Gaussian free field on $r$-regular trees. For random $r$-regular graphs of diverging size, for every fixed $r\ge3$, we show that the rescaled extremal point process of the field is asymptotically distributed, in the annealed sense, as a Poisson point process on the line with intensity $e^{-x}\,\mathrm{d}x$. The same limit behaviour is obeyed by the restriction of the GFF on $r$-regular trees to finite subsets of vertices. Our approach relies on a direct Gaussian comparison argument and precise Green function estimates.

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Lisa Hartung, Andreas Klippel, Christian Mönch. 2025-11-18. Extremes of the zero-average Gaussian Free Field on random regular graphs. https://arxiv.org/abs/2511.14026

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