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Extreme local extrema of the sine-Gordon field
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abstract
We prove that for $\beta<6\pi$ the local extremal process of the massive sine-Gordon field on the unit torus in $d=2$ converges to a Poisson point process with random intensity measure ${\rm Z}^{\mathrm{SG}}(dx) \otimes e^{-\alpha h}dh$ for some $\alpha>0$. The proof combines existing methods for the extremal process associated to the Gaussian free field, which was introduced and studied by Biskup and Louidor, and a strong coupling between the sine-Gordon field and 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.
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