We study the isoperimetric subgraphs of the infinite cluster C ∞ for supercritical bond percolation on Z d with d ≥ 3. Specifically, we consider subgraphs of C ∞ ∩ [−n, n] d having minimal open edge boundary to volume ratio. We prove a shape theorem for these subgraphs: when suitably rescaled, they converge almost surely to a translate of a deterministic shape. This deterministic shape is itself an isoperimetric set for a norm we construct. As a corollary, we obtain sharp asymptotics on a natural modification of the Cheeger constant for C ∞ ∩ [−n, n] d , settling a conjecture of Benjamini for the version of the Cheeger constant defined here. 1 arXiv:1602.05598v2 [math.PR] 27 Oct 2017 5Gibbs [32] who postulated that, in general, the asymptotic shape of the crystalline phase should minimize this surface energy. The Wulff construction furnishes this minimal shape.The spirit of Theorem 1.2 can be traced back to the work of Minlos and Sinai [44,45] from the 1960s, in which the geometric properties of phase separation in a material are rigorously studied. The first rigorous characterizations of phase separation via the Wulff construction are due independently to Dobrushin, Kotecký and Shlosman [25] in the context of the two-dimensional Ising model and to Alexander, Chayes and Chayes [2] in the context of two-dimensional bond percolation. The results of [25], valid in the low-temperature regime, were extended up to the critical temperature thanks to the work of Ioffe [37] and Ioffe and Schonmann [38].The first rigorous derivation of the Wulff construction for a genuine short-range model in three dimensions was achieved by Cerf in the context of bond percolation [15]. Analogous results for the Ising model and in higher dimensions were achieved in several substantial works of Bodineau [8,9] and Cerf and Pisztora [17,18]. The coarse graining results of Pisztora [50] played an integral role in this study of the Ising model, FK percolation and bond percolation in higher dimensions. A comprehensive survey of these results and of others can be found in Section 5.5 of Cerf's monograph [16] and in the review article of Bodineau, Ioffe and Velenik [10].In all cases, the jump to dimensions strictly larger than two has, at least so far, necessitated a shift from the uniform topology to the 1 -topology on the space of shapes (we are intentionally vague about which space we consider). Indeed, the variational problem (1.7) is not stable in d ≥ 3 when the space of shapes is equipped with the uniform topology: it is possible to construct a sequence of shapes bounded away from the optimal shape in the uniform topology, but whose surface energies tend to the optimal surface energy. This has implications at the microscopic level; to prove a uniform shape theorem in d ≥ 3 for the Cheeger optimizers, one would first have to rule out the existence of long thin filaments (as in Figure 1) in these discrete objects with high probability.6 10

We study the isoperimetric subgraphs of the giant component C n of supercritical bond percolation on the square lattice. These are subgraphs of C n having minimal edge boundary to volume ratio. In contrast to the work of [7], the edge boundary is taken only within C n instead of the full infinite cluster. The isoperimetric subgraphs are shown to converge almost surely, after rescaling, to the collection of optimizers of a continuum isoperimetric problem emerging naturally from the model. We also show that the Cheeger constant of C n scales to a deterministic constant, which is itself an isoperimetric ratio, settling a conjecture of Benjamini in dimension two.

E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. AbstractWe consider a continuous time random walk on the two-dimensional discrete torus, whose motion is governed by the discrete Gaussian free field on the corresponding box acting as a potential. More precisely, at any vertex the walk waits an exponentially distributed time with mean given by the exponential of the field and then jumps to one of its neighbors, chosen uniformly at random. We prove that throughout the low-temperature regime and at in-equilibrium timescales, the process admits a scaling limit as a spatial K-process driven by a random trapping landscape, which is explicitly related to the limiting extremal process of the field. Alternatively, the limiting process is a supercritical Liouville Brownian motion with respect to the continuum Gaussian free field on the box. This demonstrates rigorously and for the first time, as far as we know, a dynamical freezing in a spin glass system with logarithmically correlated energy levels.(1.4)We refer to K (τ ) as the K-process associated with depths τ . Given an additional sequence ξ ≡ (ξ k ) ∞ k=1 of points in V := [0, 1] 2 , we may also define the process(1.5)We refer to ξ as a sequence of trap locations and to Y (ξ,τ ) as the spatial K-process associated with trapping landscape (ξ, τ ).Next, we recall some results from the extreme value theory for the DGFF. To this end, define the structured extremal process of h N as the point process on V × R × [0, ∞) Z 2 given by η N,r := x∈V N δ x/N ⊗ δ h N,x −m N ⊗ δ (h N,x −h N,x+y )y∈B r 1 {h N,x ≥h N,x+y : y∈Br} , (1.6) EJP 23 (2018), paper 59. Page 2/31 ejp.ejpecp.org t 0 d * (f (s), g(s))ds for f, g ∈ L([0, t], V * ). Observe that under · L([0,t],V * ) (which is not a norm, despite the notation) this space is complete and separable, and that this metric generates the topology of convergence in measure on functions from [0, t] to V * . Here, the interval [0, t] is implicitly equipped with Lebesgue measure. The next theorem is the principle result of this work.Theorem A. Let β > α. Then with s N := gN 2 √ gβ (log N ) 1−3 √ gβ/4 and for any t > 0, 1 N X N (s N t) : t ∈ [0, t] =⇒ Y (β) (t) : t ∈ [0, t] as N → ∞ , (1.11) where the above weak convergence is that of random functions in L [0, t], V * .

We provide an upper bound on the number of ordered Reidemeister moves required to pass between two diagrams of the same link. This bound is in terms of the number of unordered Reidemeister moves required.Comment: 11 pages, 15 figures; fixed minor typographical erro