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.
@cite_21: Let U be a multiply connected region in R 2 with smooth boundary. Let P∈be a polyomino in ∈Z 2 approximating U as ∈ → 0. We show that, for certain boundary conditions on P∈, the height distribution on a random domino tiling (dimer covering) of P∈ is conformally invariant in the limit as ∈ tends to 0, in the sense that the distribution of heights of boundary components (or rather, the difference of the heights from their mean values) only depends on the conformal type of U. The mean height is not strictly conformally invariant but transforms analytically under conformal mappings in a simple way. The mean height and all the moments are explicitly evaluated.
@cite_13: We prove that the chordal contour lines of the discrete Gaussian free field converge to forms of SLE(4). Specifically, there is a constant λ > 0 such that when h is an interpolation of the discrete Gaussian free field on a Jordan domain—with boundary values −λ on one boundary arc and λ on the complementary arc—the zero level line of h joining the endpoints of these arcs converges to SLE(4) as the domain grows larger. If instead the boundary values are −a 0 on the complementary arc, then the convergence is to SLE(4; a λ - 1, b λ - 1), a variant of SLE(4).
The two-dimensional DGFF is particularly interesting because its continuum version (CGFF) is invariant under conformal transformations of the underlying domain. This offers a framework for analyzing scaling limits of certain critical models. For instance, the level sets of the DGFF on the triangular lattice can be linked with the Schramm-Loewner process SLE @math (Schramm and Sheffield @cite_13 ), the height function associated with domino tilings scales to the CGFF (Kenyon @cite_21 ) etc. A complication associated with the CGFF is that, by its scale-invariant nature, the field'' exists only as a random distribution on an appropriate function space.