The Percolation Transition for the Zero-Temperature Stochastic Ising Model on the Hexagonal Lattice

On the planar hexagonal lattice , we analyze the Markov process whose state (t), in , updates each site v asynchronously in continuous time t0, so that _v (t) agrees with a majority of its (three) neighbors. The initial _v (0)'s are i.i.d. with P[ _v (0)=+1]=p[0,1]. We study, both rigorously and by Monte Carlo simulation, the existence and nature of the percolation transition as t and p1/2. Denoting by ⁺(t,p) the expected size of the plus cluster containing the origin, we (1) prove that ⁺(,1/2)= and (2) study numerically critical exponents associated with the divergence of ⁺(,p) as p1/2. A detailed finite-size scaling analysis suggests that the exponents and of this t= (dependent) percolation model have the same values, 4/3 and 43/18, as standard two-dimensional independent percolation. We also present numerical evidence that the rate at which (t)() as t is exponential.     

The Percolation Signature of the Spin Glass Transition

Magnetic ordering at low temperature for Ising ferromagnets manifests itself within the associated Fortuin–Kasteleyn (FK) random cluster representation as the occurrence of a single positive density percolating network. In this paper we investigate the percolation signature for Ising spin glass ordering—both in short-range (EA) and infinite-range (SK) models—within a two-replica FK representation and also within the different Chayes–Machta–Redner two-replica graphical representation. Based on numerical studies of the ±J EA model in three dimensions and on rigorous results for the SK model, we conclude that the spin glass transition corresponds to the appearance of two percolating clusters of unequal densities.     

Approximate Lifshitz Law for the Zero-Temperature Stochastic Ising Model in any Dimension

We study the Glauber dynamics for the zero-temperature stochastic Ising model in dimension d ≥ 4 with “plus” boundary condition. Let ${\mathcal{T}_+}$ be the time needed for an hypercube of size L entirely filled with “minus” spins to become entirely “plus”. We prove that ${\mathcal{T}_+}$ is O(L ²(log L) ^c ) for some constant c, not depending on the dimension. This brings further rigorous justification for the so-called “Lifshitz law” ${\mathcal{T}_{+} = O(L^{2})}$ (Fischer and Huse in Phys Rev B 35:6841–6848, 1987; Lifshitz in Sov Phys JETP 15:939–942, 1962) conjectured on heuristic grounds. The key point of our proof is to use the detailed knowledge that we have on the three-dimensional problem: results for fluctuation of monotone interfaces at equilibrium and mixing time for monotone interfaces dynamics extracted from Caputo et al. (Comm Pure Appl Math 64:778–831, 2011) to get the result in higher dimension.     

A New Proof of the Sharpness of the Phase Transition for Bernoulli Percolation and the Ising Model

We provide a new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. The proof applies to infinite-range models on arbitrary locally finite transitive infinite graphs. For Bernoulli percolation, we prove finiteness of the susceptibility in the subcritical regime \({\beta < \beta_c}\), and the mean-field lower bound \({\mathbb{P}_\beta[0\longleftrightarrow \infty ]\ge (\beta-\beta_c)/\beta}\) for \({\beta > \beta_c}\). For finite-range models, we also prove that for any \({\beta < \beta_c}\), the probability of an open path from the origin to distance n decays exponentially fast in n. For the Ising model, we prove finiteness of the susceptibility for \({\beta < \beta_c}\), and the mean-field lower bound \({\langle \sigma_0\rangle_\beta^+\ge \sqrt{(\beta^2-\beta_c^2)/\beta^2}}\) for \({\beta > \beta_c}\). For finite-range models, we also prove that the two-point correlation functions decay exponentially fast in the distance for \({\beta < \beta_c}\).     

Phase Transition and Uniqueness of Levelset Percolation

The main purpose of this paper is to introduce and establish basic results of a natural extension of the classical Boolean percolation model (also known as the Gilbert disc model). We replace the balls of that model by a positive non-increasing attenuation function \(l:(0,\infty ) \rightarrow [0,\infty )\) to create the random field \(\Psi (y)=\sum _{x\in \eta }l(|x-y|),\) where \(\eta \) is a homogeneous Poisson process in \({\mathbb {R}}^d.\) The field \(\Psi \) is then a random potential field with infinite range dependencies whenever the support of the function l is unbounded. In particular, we study the level sets \(\Psi _{\ge h}(y)\) containing the points \(y\in {\mathbb {R}}^d\) such that \(\Psi (y)\ge h.\) In the case where l has unbounded support, we give, for any \(d\ge 2,\) a necessary and sufficient condition on l for \(\Psi _{\ge h}(y)\) to have a percolative phase transition as a function of h. We also prove that when l is continuous then so is \(\Psi \) almost surely. Moreover, in this case and for \(d=2,\) we prove uniqueness of the infinite component of \(\Psi _{\ge h}\) when such exists, and we also show that the so-called percolation function is continuous below the critical value \(h_c\).     

Phase Transition and Level-Set Percolation for the Gaussian Free Field

We consider level-set percolation for the Gaussian free field on ${\mathbb{Z}^{d}}$ , 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 Bricmont et al. (J Stat Phys 48(5/6):1249–1268, 1987). 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.     

Percolation in a Voronoi Competition-Growth Model

We study a model in which two entities (e.g., plant species, political ideas,...) compete for space on a plane, starting from randomly distributed seeds and growing deterministically at possibly different rates. An entity which forms an infinite cluster is considered to dominate over the other (which then cannot percolate). We analyze the occurrence of such a form of domination in situations in which one entity starts from a much larger density of seeds than the other one, but the latter one grows at a much faster rate than the former one. The model studied here generalizes the problem of Voronoi percolation.     

自发磁化渗流模型的可视化演示实验

经过多年的发展,统计物理对自发磁化进行了深入研究并取得丰硕成果。但是对该理论,很多学生难以建立相应物理图像。借助自发磁化的渗流模型和Java语言,我们将自发磁化过程中铁磁物质磁性有无与系统微观形貌上所定义集团连通与否这一定量关系动态直观地展示给学生,从而达到增加感性认识,提高教学效果的目的。     

The Boundary Structure of Zero-Temperature Driven Hard Spheres

We study the fundamental problem of two gas species whose molecules collide as hard spheres in the presence of a flat boundary and with dependence on only one space dimension. More specifically the steady linear problem considered is the one arising when the second gas dominates as a flow moving towards the boundary with constant microscopic velocity (and hence zero temperature). Theboundary condition adopted consists of prescribing the outgoing velocity distribution of the firstgas at the boundary. It is discovered that the presence of the boundary under general assumptions on the outgoing distribution ensures the convergence of a series of path integrals resulting in a convenient representation for the distribution of the velocities of the molecules returning at the boundary.     

The High Temperature Ising Model on the Triangular Lattice is a Critical Bernoulli Percolation Model

We define a new percolation model by generalising the FK representation of the Ising model, and show that on the triangular lattice and at high temperatures, the critical point in the new model corresponds to the Ising model. Since the new model can be viewed as Bernoulli percolation on a random graph, our result makes an explicit connection between Ising percolation and critical Bernoulli percolation, and gives a new justification of the conjecture that the high temperature Ising model on the triangular lattice is in the same universality class as Bernoulli percolation.     

The Influence of the Percolation Transition on the Electric Conductive and Optical Properties of Ultrathin Metallic Films

Physics of the Solid State - The results of the study of the peculiarities of changes in the electrophysical, optical, and plasmonic properties of ultrathin metallic films during the percolation...     

Energy of Long-Lifetime Configurations in Zero-Temperature Dynamics

Using Monte Carlo method with zero-temperature dynamics, we investigate energy evolution of Ising spin configuration on a square lattice. The energies of some configurations exhibit long duration before those configurations reach the final state -- ground state or frozen stripe state. For ground-state dynamical realization, the duration occurs when the energy per spin is 4/L, where L is the lattice size. For stripe-state dynamical realization, the energy is slightly higher than 2/L when the duration appears in the last evolution stage. In addition, it is found that the average energy per spin in final state is approximately 2/3L.     

On the Zero-Temperature Limit of Gibbs States

We exhibit Lipschitz (and hence Hölder) potentials on the full shift ${\{0,1\}^{\mathbb{N}}}We exhibit Lipschitz (and hence H?lder) potentials on the full shift {0,1}^\mathbbN{\{0,1\}^{\mathbb{N}}} such that the associated Gibbs measures fail to converge as the temperature goes to zero. Thus there are “exponentially decaying” interactions on the configuration space {0,1}^{\mathbb Z}{\{0,1\}^{\mathbb Z}} for which the zero-temperature limit of the associated Gibbs measures does not exist. In higher dimension, namely on the configuration space {0,1}^\mathbbZd{\{0,1\}^{\mathbb{Z}^{d}}}, d ≥ 3, we show that this non-convergence behavior can occur for the equilibrium states of finite-range interactions, that is, for locally constant potentials.     

Local Neighbourhoods for First-Passage Percolation on the Configuration Model

We consider first-passage percolation on the configuration model. Once the network has been generated each edge is assigned an i.i.d. weight modeling the passage time of a message along this edge. Then independently two vertices are chosen uniformly at random, a sender and a recipient, and all edges along the geodesic connecting the two vertices are coloured in red (in the case that both vertices are in the same component). In this article we prove local limit theorems for the coloured graph around the recipient in the spirit of Benjamini and Schramm. We consider the explosive regime, in which case the random distances are of finite order, and the Malthusian regime, in which case the random distances are of logarithmic order.     

Slab Percolation and Phase Transitions for the Ising Model

We prove, using the random-cluster model, a strict inequality between site percolation and magnetization in the region of phase transition for the d-dimensional Ising model, thus improving a result of [5]. We extend this result also at the case of two plane lattices (slabs) and give a characterization of phase transition in this case. The general case of N slabs, with N an arbitrary positive integer, is partially solved and it is used to show that this characterization holds in the case of three slabs with periodic boundary conditions. AMS classification: 60K35, 82B20, 82A25     

Four Types of Percolation Transitions in the Cluster Aggregation Network Model

We study the percolation transition in a one-species cluster aggregation network model, in which the parameter α describes the suppression on the cluster sizes. It is found that the model can exhibit four types of percolation transitions, two continuous percolation transitions and two discontinuous ones. Continuous and discontinuous percolation transitions can be distinguished from each other by the largest single jump. Two types of continuous percolation transitions show different behaviors in the time gap. Two types of discontinuous percolation transitions are different in the time evolution of the cluster size distribution. Moreover, we also find that the time gap may also be a measure to distinguish different discontinuous percolations in this model.     

Truncated Connectivities in a Highly Supercritical Anisotropic Percolation Model

We consider an anisotropic bond percolation model on $\mathbb{Z}^{2}$ , with p=(p _h ,p _v )∈[0,1]², p _v >p _h , and declare each horizontal (respectively vertical) edge of $\mathbb{Z}^{2}$ to be open with probability p _h (respectively p _v ), and otherwise closed, independently of all other edges. Let $x=(x_{1},x_{2}) \in\mathbb{Z}^{2}$ with 0<x ₁<x ₂, and $x'=(x_{2},x_{1})\in\mathbb{Z}^{2}$ . It is natural to ask how the two point connectivity function $\mathbb{P}_{\mathbf{p}}(\{0\leftrightarrow x\})$ behaves, and whether anisotropy in percolation probabilities implies the strict inequality $\mathbb{P}_{\mathbf{p}}(\{0\leftrightarrow x\})>\mathbb{P}_{\mathbf {p}}(\{0\leftrightarrow x'\})$ . In this note we give an affirmative answer in the highly supercritical regime.     

Percolation and the Existence of a Soft Phase in the Classical Heisenberg Model

We present the results of a numerical investigation of percolation properties in a version of the classical Heisenberg model. In particular we study the percolation properties of the subsets of the lattice corresponding to equatorial strips of the target manifold ². As shown by us several years ago, this is relevant for the existence of a massless phase of the model. Our investigation yields strong evidence that such a massless phase does indeed exits. It is further shown that this result implies lack of asymptotic freedom in the massive continuum limit. A heuristic estimate of the transition temperature is given which is consistent with the numerical data.