The extremal process of centered square-root local time on the leaves of a regular tree converges to a decorated Poisson point process with the same cluster law as the Gaussian Free Field.
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Extremal process of the local time of simple random walk on a regular tree
The extremal process of centered square-root local time on the leaves of a regular tree converges to a decorated Poisson point process with the same cluster law as the Gaussian Free Field.