1202.5172 (Pierre-François Rodriguez et al.)

Phase transition and level-set percolation for the Gaussian free field    [PDF]

Pierre-François Rodriguez, Alain-Sol Sznitman

We consider level-set percolation for the Gaussian free field on Z^d, 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.

View original: http://arxiv.org/abs/1202.5172