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Disconnection and level-set percolation for the Gaussian free field. (English) Zbl 1337.60246
The paper under review is a study of the level-set percolation of the Gaussian free field on \(Z^d\) for \(d\geq 3\) and it aims to derive the upper and lower bounds on the probability that a box of large side-length gets disconnected from the boundary of a larger homothetic box by the excursion-set of the Gaussian free field below the level \(\alpha\).
By results of J. Bricmont et al. [J. Stat. Phys. 48, No. 5–6, 1249–1268 (1987; Zbl 0962.82520)] and P.-F. Rodriguez and A.-S. Sznitman [Commun. Math. Phys. 320, No. 2, 571–601 (2013; Zbl 1269.82028)], there is a critical value \(0\leq h^* < \infty\) such that the level set \(E^{\geq \alpha}=\{x\in Z^d: \phi_x\geq \alpha\}\) has only finite connected components a.s. for \(\alpha > h_*\) and the level set \(E^{\geq \alpha}\) has a unique infinite connected component a.s. for \(\alpha < h_*\), where \(\phi = (\phi_x)_{x\in Z^d}\) is the canonical process in the canonical law of the discrete Gaussian free field. There is another critical value \(h_{**} =\inf \{\alpha \in \mathbb{R}: \lim_L \inf \operatorname{P}[B_L \overset{\geq \alpha}{\longleftrightarrow} \partial B_{2L}] =0\}\). Define the event \(A_N\) as the complement of the event \(B_N \overset{\geq \alpha}{\longleftrightarrow} \partial S_N=\{x\in Z^d: |x|_{\infty} =|MN|\}\). In the strongly non-percolative regime \(\alpha > h_{**}\) of the excursion set \(E^{\geq \alpha}\), one has \(\lim_N \operatorname{P}[A_N]=1\).
The main result of the paper is Theorem 5.5 that there is a critical value \(\overline{h} = \sup \{h\in\mathbb R: \text{for all }h>\alpha > \beta\), \(\phi\) strongly percolates at levels \(\alpha, \beta\)

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G60 Random fields
60G15 Gaussian processes
60G50 Sums of independent random variables; random walks
60F10 Large deviations
82B43 Percolation
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