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Phase Transition and Level-Set Percolation for the Gaussian Free Field

Rodriguez, Pierre-François ; Sznitman, Alain-Sol

In: Communications in Mathematical Physics, 2013, vol. 320, no. 2, p. 571-601

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    Title
    • Phase Transition and Level-Set Percolation for the Gaussian Free Field
    Author
    • Rodriguez, Pierre-François. Departement Mathematik, ETH Zürich, 8092, Zürich, Switzerland
    • Sznitman, Alain-Sol. Departement Mathematik, ETH Zürich, 8092, Zürich, Switzerland
    Document Type
    • Postprint
    Language
    • English
    Published in
    • Communications in Mathematical Physics, 2013, vol. 320, no. 2, p. 571-601. Springer-Verlag
    Other Version
    Classification
    • Mathematics
    OAI-PMH Identifier
    • oai:doc.rero.ch:312161
    Summary
    We consider level-set percolation for the Gaussian free field on $${\mathbb{Z}^{d}}$$ , d ≥ 3, and prove that, as h varies, there is a non-trivial percolation phase transition of the excursion set above level h for all dimensions d ≥ 3. So far, it was known that the corresponding critical level h *(d) satisfies h *(d) ≥ 0 for all d ≥ 3 and that h *(3) is finite, see Bricmont etal. (J Stat Phys 48(5/6):1249-1268, 1987). We prove here that h *(d) is finite for all d ≥ 3. In fact, we introduce a second critical parameter h ** ≥ h *, show that h **(d) is finite for all d ≥ 3, and that the connectivity function of the excursion set above level h has stretched exponential decay for all h> h **. Finally, we prove that h * is strictly positive in high dimension. It remains open whether h * and h ** actually coincide and whether h * > 0 for all d ≥ 3