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 [2]. 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.to a thick two-dimensional slab percolates, see above (0.12). Let us however point out that, by a result of [7] (see p. 1151 therein), the restriction of E ≥0 ϕ to Z 2 (viewed as a subset of Z d ), and, a fortiori, the restriction of E ≥h ϕ to Z 2 , when h is positive, do only contain finite connected components: excursion sets above any non-negative level do not percolate in planes. We refer to Remark 3.6 1) for more on this.We now comment on the proofs. We begin with (0.8). The key ingredient is a certain (static) renormalization scheme very similar to the one developed in Section 2 of [21] for the problem of percolation of the vacant set left by random interlacements (for a precise definition of this model, see [23], Section 1; we merely note that the two "corresponding" quantities are E ≥h ϕ and

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.

We study continuous-time (variable speed) random walks in random environments on Z d , d ≥ 2, where, at time t, the walk at x jumps across edge (x, y) at time-dependent rate a t (x, y). The rates, which we assume stationary and ergodic with respect to spacetime shifts, are symmetric and bounded but possibly degenerate in the sense that the total jump rate from a vertex may vanish over finite intervals of time. We formulate conditions on the environment under which the law of diffusively-scaled random walk path tends to Brownian motion for almost every sample of the rates. The proofs invoke Moser iteration to prove sublinearity of the corrector in pointwise sense; a key additional input is a conversion of certain weighted energy norms to ordinary ones. Our conclusions apply to random walks on dynamical bond percolation and interacting particle systems as well as to random walks arising from the Helffer-Sjöstrand representation of gradient models with certain non-strictly convex potentials.

We investigate level sets of the Gaussian free field on continuous transient metric graphs $$\widetilde{{\mathcal {G}}}$$ G ~ and study the capacity of its level set clusters. We prove, without any further assumption on the base graph $${\mathcal {G}}$$ G , that the capacity of sign clusters on $$\widetilde{{\mathcal {G}}}$$ G ~ is finite almost surely. This leads to a new and effective criterion to determine whether the sign clusters of the free field on $$\widetilde{{\mathcal {G}}}$$ G ~ are bounded or not. It also elucidates why the critical parameter for percolation of level sets on $$\widetilde{{\mathcal {G}}}$$ G ~ vanishes in most instances in the massless case and establishes the continuity of this phase transition in a wide range of cases, including all vertex-transitive graphs. When the sign clusters on $$\widetilde{{\mathcal {G}}}$$ G ~ do not percolate, we further determine by means of isomorphism theory the exact law of the capacity of compact clusters at any height. Specifically, we derive this law from an extension of Sznitman’s refinement of Lupu’s recent isomorphism theorem relating the free field and random interlacements, proved along the way, and which holds under the sole assumption that sign clusters on $$\widetilde{{\mathcal {G}}}$$ G ~ are bounded. Finally, we show that the law of the cluster capacity functionals obtained in this way actually characterizes the isomorphism theorem, i.e. the two are equivalent.

Simulating self-avoiding walks in bounded domains J. Math. Phys. 53, 095219 (2012) Stability and clustering of self-similar solutions of aggregation equations J. Math. Phys. 53, 115610 (2012) The scaling limit of the energy correlations in non-integrable Ising models J. Math. Phys. 53, 095214 (2012) Location of the Lee-Yang zeros and absence of phase transitions in some Ising spin systems For lattice systems of statistical mechanics satisfying a Lee-Yang property (i.e., for which the Lee-Yang circle theorem holds), we present a simple proof of analyticity of (connected) correlations as functions of an external magnetic field h, for Reh = 0. A survey of models known to have the Lee-Yang property is given. We conclude by describing various applications of the aforementioned analyticity in h. C 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4749391] This note is dedicated to Elliott Lieb, mentor and friend, on the occasion of his eightieth birthday.