2012/02/29 by Pierre-François Rodriguez, Alain-Sol Sznitman · 5 citations
Mathematics · Physics and Astronomy · #math.PR #math-ph #math.MP #msc:60K35 #msc:60G15 #msc:60G60 #msc:82B43
paper · pdf · doi:10.1007/s00220-012-1649-y
published as Commun. Math. Phys. 320, 571-601 (2013) · 29 pages, 1 figure, accepted for publication in Communications in Mathematical Physics
arxiv created 2013/01/11 · arxiv updated 2013/07/23
We consider level-set percolation for the Gaussian free field on Zd, with d bigger or equal to 3, and prove that there is a non-trivial critical level h_* such that for h > h_*, the excursion set above level h does not percolate, and for h < h_*, the excursion set does percolate. It is known from the work of Bricmont-Lebowitz-Maes that h_* is non-negative for all d bigger or equal to 3, and finite, when d=3. We prove here that h_* is finite for all d bigger or equal to 3. In fact, we introduce a second critical parameter h_**, which is bigger or equal to h_*. We show that h_** is finite for all d bigger or equal to 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_* > 0 in high dimension. It remains open whether h_* and h_** actually coincide, and whether h_* > 0 for all d bigger or equal to 3.