We consider level-sets of the Gaussian free field on Z d , for d ≥ 3, above a given realvalued height parameter h. As h varies, this defines a canonical percolation model with strong, algebraically decaying correlations. We prove that three natural critical parameters associated to this model, namely h * * (d), h * (d) and h(d), respectively describing a wellordered subcritical phase, the emergence of an infinite cluster, and the onset of a local uniqueness regime in the supercritical phase, actually coincide, i.e. h * * (d) = h * (d) = h(d) for any d ≥ 3. At the core of our proof lies a new interpolation scheme aimed at integrating out the long-range dependence of the Gaussian free field. The successful implementation of this strategy relies extensively on certain novel renormalization techniques, in particular to control so-called large-field effects. This approach opens the way to a complete understanding of the off-critical phases of strongly correlated percolation models.

Given a planar continuum Gaussian free field h𝒰 in a domain 𝒰 with Dirichlet boundary condition and any δ > 0, we let {hδscriptU(v):v∈U} be a real‐valued smooth Gaussian process where hδscriptU(v) is the average of h𝒰 along a circle of radius δ with center v. For γ > 0, we study the Liouville first‐passage percolation (in scale δ), i.e., the shortest path distance in 𝒰 where the weight of each path P is given by ∫PeγhδUfalse(zfalse)|dz|. We show that the distance between two typical points is O(δc*γ4/3/logγ−1) for all sufficiently small but fixed γ > 0 and some constant c* > 0. In addition, we obtain similar upper bounds on the Liouville first‐passage percolation for discrete Gaussian free fields, as well as the Liouville graph distance, which roughly speaking is the minimal number of euclidean balls with comparable Liouville quantum gravity measure whose union contains a continuous path between two endpoints. Our results contradict some reasonable interpretations of Watabiki's prediction (1993) on the random distance of Liouville quantum gravity at high temperatures.© 2019 Wiley Periodicals, Inc.

The truncated two-point function of the ferromagnetic Ising model on Z d (d ≥ 3) in its pure phases is proven to decay exponentially fast throughout the ordered regime (β > β c and h = 0). Together with the previously known results, this implies that the exponential clustering property holds throughout the model's phase diagram except for the critical point: (β, h) = (β c , 0).

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We consider the branching random walk {R N z : z ∈ V N } with Gaussian increments indexed over a two-dimensional box V N of side length N , and we study the first passage percolation where each vertex is assigned weight e γR N z for γ > 0. We show that for γ > 0 sufficiently small but fixed, the expected FPP distance between the left and right boundaries is at most O(N 1−γ 2 /10 ).