Dobrushin (1972) showed that the interface of a 3D Ising model with minus boundary conditions above the @math -plane and plus below is rigid (has @math -fluctuations) at every sufficiently low temperature. Since then, basic features of this interface -- such as the asymptotics of its maximum -- were only identified in more tractable random surface models that approximate the Ising interface at low temperatures, e.g., for the (2+1)D Solid-On-Solid model. Here we study the large deviations of the interface of the 3D Ising model in a cube of side-length @math with Dobrushin's boundary conditions, and in particular obtain a law of large numbers for @math , its maximum: if the inverse-temperature @math is large enough, then @math as @math , in probability, where @math is given by a large deviation rate in infinite volume. We further show that, on the large deviation event that the interface connects the origin to height @math , it consists of a 1D spine that behaves like a random walk, in that it decomposes into a linear (in @math ) number of asymptotically-stationary weakly-dependent increments that have exponential tails. As the number @math of increments diverges, properties of the interface such as its surface area, volume, and the location of its tip, all obey CLTs with variances linear in @math . These results generalize to every dimension @math .
While cluster expansion only converges at sufficiently large @math , it is natural to ask if the rigidity of the interface, and our new results, hold for all @math . This is not believed to be the case, as the Ising model is widely believed to undergo a for @math (and no other dimension): Much like the SOS and DG approximations, which exhibit phase transitions in @math ---whereby they roughen and resemble the discrete Gaussian free field @cite_7 @cite_33 for small @math ---it is conjectured that for the 3D Ising model there exists a point @math such that, for @math , the model has long-range order, yet the typical fluctuations of its horizontal interface diverge with @math ; proving this transition is a longstanding open problem (see, e.g, @cite_38 @cite_30 ).
@cite_30 @cite_38 @cite_33 @cite_7
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We study an evolving spatial network in which sequentially arriving vertices are joined to existing vertices at random according to a rule that combines preference according to degree with preference according to spatial proximity. We investigate phase transitions in graph structure as the relative weighting of these two components of the attachment rule is varied. Previous work of one of the authors showed that when the geometric component is weak, the limiting degree sequence of the resulting graph coincides with that of the standard Barab 'asi--Albert preferential attachment model. We show that at the other extreme, in the case of a sufficiently strong geometric component, the limiting degree sequence coincides with that of a purely geometric model, the on-line nearest-neighbour graph, which is of interest in its own right and for which we prove some extensions of known results. We also show the presence of an intermediate regime, in which the behaviour differs significantly from both the on-line nearest-neighbour graph and the Barab 'asi--Albert model; in this regime, we obtain a stretched exponential upper bound on the degree sequence. Our results lend some mathematical support to simulation studies of Manna and Sen, while proving that the power law to stretched exponential phase transition occurs at a different point from the one conjectured by those authors. This is the second in a series of papers in which we derive a @math -expansion for the two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. In our previous paper, we computed the @math -limit of the leading order energy, which yields the averaged behavior for almost minimizers, namely that the density of droplets should be uniform. Here we go to the next order and derive a next order @math -limit energy, which is exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a limiting interaction energy for vortices in the magnetic Ginzburg-Landau model. The derivation is based on the abstract scheme of Sandier-Serfaty that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Without thus appealing to the Euler-Lagrange equation, we establish for all configurations which have "almost minimal energy" the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense. Via a kind of @math -equivalence, the obtained results also yield an expansion of the minimal energy for the original Ohta-Kawasaki energy. This leads to expecting to see triangular lattices of droplets as energy minimizers. We establish some scaling limits for a model of planar aggregation. The model is described by the composition of a sequence of independent and identically distributed random conformal maps, each corresponding to the addition of one particle. We study the limit of small particle size and rapid aggregation. The process of growing clusters converges, in the sense of Caratheodory, to an inflating disc. A more refined analysis reveals, within the cluster, a tree structure of branching fingers, whose radial component increases deterministically with time. The arguments of any finite sample of fingers, tracked inwards, perform coalescing Brownian motions. The arguments of any finite sample of gaps between the fingers, tracked outwards, also perform coalescing Brownian motions. These properties are closely related to the evolution of harmonic measure on the boundary of the cluster, which is shown to converge to the Brownian web. We consider a Glauber dynamics reversible with respect to the two-dimensional Ising model in a finite square of sideL, in the absence of an external field and at large inverse temperature β. We first consider the gap in the spectrum of the generator of the dynamics in two different cases: with plus and open boundary conditions. We prove that, when the symmetry under global spin flip is broken by the boundary conditions, the gap is much larger than the case in which the symmetry is present. For this latter we compute exactly the asymptotics of −(1 βL) log(gap) asL→∞ and show that it coincides with the surface tension along one of the coordinate axes. As a consequence we are able to study quite precisely the large deviations in time of the magnetization and to obtain an upper bound on the spin-spin time correlation in the infinite-volume plus phase. Our results establish a connection between the dynamical large deviations and those of the equilibrium Gibbs measure studied by Shlosman in the framework of the rigorous description of the Wulff shape for the Ising model. Finally we show that, in the case of open boundary conditions, it is possible to rescale the time withL in such a way that, asL→∞, the finite-dimensional distributions of the time-rescaled magnetization converge to those of a symmetric continuous-time Markov chain on the two-state space −m*(β),m*(β) ,m*(β) being the spontaneous magnetization. Our methods rely upon a novel combination of techniques for bounding from below the gap of symmetric Markov chains on complicated graphs, developed by Jerrum and Sinclair in their Markov chain approach to hard computational problems, and the idea of introducing “block Glauber dynamics” instead of the standard single-site dynamics, in order to put in evidence more effectively the effect of the boundary conditions in the approach to equilibrium.