We consider the discrete Gaussian Free Field in a square box in ( Z ^2 ) of side length N with zero boundary conditions and study the joint law of its properly-centered extreme values (h) and their scaled spatial positions (x) in the limit as ( N ). Restricting attention to extreme local maxima, i.e., the extreme points that are maximal in an r N -neighborhood thereof, we prove that the associated process tends, whenever ( r_N ) and ( r_N N 0 ), to a Poisson point process with intensity measure ( Z ( dx) e ^ - h d h ), where ( := 2 g ) with g: = 2 π and where Z(dx) is a random Borel measure on [0, 1]2. In particular, this yields an integral representation of the law of the absolute maximum, similar to that found in the context of Branching Brownian Motion. We give evidence that the random measure Z is a version of the derivative martingale associated with the continuum Gaussian Free Field.
Broadening our discussion to subjects that are not primarily concerned with the behavior of the extreme points, let us mention also the work of Daviaud @cite_8 who studied the size of the set where the field exceeds a constant times @math . His principal result is that, for @math , and @math in probability. The extreme level sets (at levels of order @math ) thus exhibit a non-trivial fractal structure. The reader should notice the striking similarity with the level sets for @math independent Gaussians with variance @math .
@cite_8
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We consider level-set percolation for the Gaussian free field on ( 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 (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.