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1.
The scaling laws of Edwards et al. for cluster fragmentation at the two-dimensional percolation threshold, recently confirmed also in high dimensions, explain the Alexandrowicz observation that the Becker–Döring equation of classical nucleation theory and its generalization by Katz, Saltsburg, and Reiss fail right at the critical point.  相似文献   

2.
The Kac-Ward formula allows to compute the Ising partition function on any finite graph G from the determinant of 22g matrices, where g is the genus of a surface in which G embeds. We show that in the case of isoradially embedded graphs with critical weights, these determinants have quite remarkable properties. First of all, they satisfy some generalized Kramers-Wannier duality: there is an explicit equality relating the determinants associated to a graph and to its dual graph. Also, they are proportional to the determinants of the discrete critical Laplacians on the graph G, exactly when the genus g is zero or one. Finally, they share several formal properties with the Ray-Singer ${\overline{\partial}}$ -torsions of the Riemann surface in which G embeds.  相似文献   

3.
Journal of Experimental and Theoretical Physics - In this paper, we review recent results on sample-to-sample fluctuations in a critical Ising model with quenched random ferromagnetic couplings. In...  相似文献   

4.
The Ising model is studied in the fermionicformulation of the stochastic quantization. An exactstochastic equation is given for D = 2 and 3 and in aHartree approximation a method is developed for treating the two-point correlation functions.  相似文献   

5.
Adapting the recent argument of Aizenman, Duminil-Copin and Sidoravicius for the classical Ising model, it is shown here that the magnetization in the transverse-field Ising model vanishes at the critical point. The proof applies to the ground state in dimension d ≥ 2 and to positive-temperature states in dimension d ≥ 3, and relies on graphical representations as well as an infrared bound.  相似文献   

6.
The character of critical behavior in physical systems depends on the range of interactions. In the limit of infinite range of the interactions, systems will exhibit mean-field critical behavior, i.e., critical behavior not affected by fluctuations of the order parameter. If the interaction range is finite, the critical behavior asymptotically close to the critical point is determined by fluctuations and the actual critical behavior depends on the particular universality class. A variety of systems, including fluids and anisotropic ferromagnets, belongs to the three-dimensional Ising universality class. Recent numerical studies of Ising models with different interaction ranges have revealed a spectacular crossover between the asymptotic fluctuation-induced critical behavior and mean-field-type critical behavior. In this work, we compare these numerical results with a crossover Landau model based on renormalization-group matching. For this purpose we consider an application of the crossover Landau model to the three-dimensional Ising model without fitting to any adjustable parameters. The crossover behavior of the critical susceptibility and of the order parameter is analyzed over a broad range (ten orders) of the scaled distance to the critical temperature. The dependence of the coupling constant on the interaction range, governing the crossover critical behavior, is discussed.  相似文献   

7.
We introduce a new version of discrete holomorphic observables for the critical planar Ising model. These observables are holomorphic spinors defined on double covers of the original multiply connected domain. We compute their scaling limits, and show their relation to the ratios of spin correlations, thus providing a rigorous proof to a number of formulae for those ratios predicted by CFT arguments.  相似文献   

8.
Fisher established an explicit correspondence between the 2-dimensional Ising model defined on a graph G and the dimer model defined on a decorated version ${\mathcal{G}}$ of this graph (Fisher in J Math Phys 7:1776–1781, 1966). In this paper we explicitly relate the dimer model associated to the critical Ising model and critical cycle rooted spanning forests (CRSFs). This relation is established through characteristic polynomials, whose definition only depends on the respective fundamental domains, and which encode the combinatorics of the model. We first show a matrix-tree type theorem establishing that the dimer characteristic polynomial counts CRSFs of the decorated fundamental domain ${\mathcal{G}_1}$ . Our main result consists in explicitly constructing CRSFs of ${\mathcal{G}_1}$ counted by the dimer characteristic polynomial, from CRSFs of G 1, where edges are assigned Kenyon’s critical weight function (Kenyon in Invent Math 150(2):409–439, 2002); thus proving a relation on the level of configurations between two well known 2-dimensional critical models.  相似文献   

9.
We consider the Ising spin 1/2 model on arbitrary pure Husimi lattices. An effective representation for the recursion relations is found which allows to write the general solution of the model in an fluent unified way for all pure Husimi lattices. In this respect, explicit expressions for the spontaneous magnetization, for the susceptibility, for the free energy, and for the specific heat are found. Besides, it is shown that this representation allows also to determine exactly the position of the critical temperature on arbitrary pure Husimi lattice. It is found that the critical temperatures for all pure Husimi lattices are driven by a single polynomial equation with coefficients given by parameters that uniquely describe the lattices.  相似文献   

10.
In this paper, the antiferromagnetic Ising model with ferromagnetic long-range interaction is modeled by the Monte Carlo method. The case of ferromagnetic long-range forces decreasing by a power law is considered. The dependence of the phase-transition temperature on long-range interaction parameters is obtained. The phase diagram was constructed at different values of long-range interaction parameters. Conditions for the existence of the frustrated state of the system were revealed.  相似文献   

11.
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.  相似文献   

12.
We present a rigorous determination of the critical value of the ground-state quantum Ising model in a transverse field, on a class of planar graphs which we call star-like. These include the junction of several copies of ℤ at a single point. Our approach is to use the graphical, or fk-, representation of the model, and the probabilistic and geometric tools associated with it. This research was carried out during the author’s Ph.D. studentship at the University of Cambridge, UK, and the Royal Institute of Technology (KTH), Sweden. The author gratefully acknowledges funding from KTH during this period. The author would also like to thank Riddarhuset, Stockholm, for generous support during his studies.  相似文献   

13.
We study the formation/dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence. Specifically, we consider the 2D Ising model in volumes of size L 2 , inverse temperature > c and overall magnetization conditioned to take the value m L 2 –2m v L , where c –1 is the critical temperature, m =m () is the spontaneous magnetization and v L is a sequence of positive numbers. We find that the critical scaling for droplet formation/dissolution is when v L 3/2 L –2 tends to a definite limit. Specifically, we identify a dimensionless parameter , proportional to this limit, a non-trivial critical value c and a function such that the following holds: For < c , there are no droplets beyond log L scale, while for > c , there is a single, Wulff-shaped droplet containing a fraction c =2/3 of the magnetization deficit and there are no other droplets beyond the scale of log L. Moreover, and are related via a universal equation that apparently is independent of the details of the system.  相似文献   

14.
The ferromagnetic Ising model without external field on an infinite Lorentzian triangulation sampled from the uniform distribution is considered. We prove uniqueness of the Gibbs measure in the high temperature region and coexistence of at least two Gibbs measures at low temperature. The proofs are based on the disagreement percolation method and on a variant of the Peierls contour method. The critical temperature is shown to be constant a.s.  相似文献   

15.
16.
Critical relaxation from the low-temperature ordered state of the three-dimensional fully frustrated Ising model on a simple cubic lattice is studied by the short-time dynamics method. Cubic systems with periodic boundary conditions and linear sizes of L = 32, 64, 96, and 128 in each crystallographic direction are studied. Calculations were carried out by the Monte Carlo method using the standard Metropolis algorithm. The static critical exponents for the magnetization and correlation radius and the dynamic critical exponents are calculated.  相似文献   

17.
Within the massive field-theoretic renormalization-group approach the expressions for the and functions of the anisotropic mn-vector model are obtained for general space dimension d in three-loop approximation. Resumming corresponding asymptotic series, critical exponents for the case of the weakly diluted quenched Ising model (m = 1, n = 0), as well as estimates for the marginal order parameter component number m c of the weakly diluted quenched m-vector model, are calculated as functions of d in the region 2 d < 4. Conclusions concerning the effectiveness of different resummation techniques are drawn.  相似文献   

18.
In Giardinà et al. (ALEA Lat Am J Probab Math Stat 13(1):121–161, 2016), the authors have defined an annealed Ising model on random graphs and proved limit theorems for the magnetization of this model on some random graphs including random 2-regular graphs. Then in Can (Annealed limit theorems for the Ising model on random regular graphs, arXiv:1701.08639, 2017), we generalized their results to the class of all random regular graphs. In this paper, we study the critical behavior of this model. In particular, we determine the critical exponents and prove a non standard limit theorem stating that the magnetization scaled by \(n^{3/4}\) converges to a specific random variable, with n the number of vertices of random regular graphs.  相似文献   

19.
20.
The saddle point equation of Ginzburg-Landau Hamiltonian for the diluted Ising model is developed. The ground state is solved numerically in two dimensions. The result is partly explained by the coarse-grained approximation.  相似文献   

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