Percolation points and critical point in the Ising model

Rigorous inequalities are proved, which relate percolation probability, mean cluster size and pair connectedness respectively with magnetization, susceptibility and pair correlation function in ferromagnetic Ising models. In two dimensions the critical point is shown to be a percolation point, while in three dimensions this is not true.

[1]  J. W. Essam,et al.  Exact Critical Percolation Probabilities for Site and Bond Problems in Two Dimensions , 1964 .

[2]  T. Lubensky.  Critical properties of random-spin models from the ∊ expansion , 1975 .

[3]  C. Fortuin,et al.  Correlation inequalities on some partially ordered sets , 1971 .

[4]  J. Hammersley,et al.  Percolation Processes and Related Topics , 1963 .

[5]  A. Bishop.  A Use of Bethe Lattice Solutions for Percolation-Constrained Phenomena , 1975 .

[6]  R. A. Muse,et al.  On randomly dilute ferromagnetic models , 1972 .

[7]  Michael E. Fisher,et al.  Theory of Critical-Point Scattering and Correlations. I. The Ising Model , 1967 .

[8]  Percolation Theory and the Ising Model for a Class of Triangular Cactii , 1974 .

[9]  A. Coniglio.  Percolation problems and phase transitions , 1975 .

[10]  L. Reatto,et al.  Droplet model equation of state and correlation function for Ising and Heisenberg models , 1972 .

[11]  R. Stinchcombe.  Conductivity and spin-wave stiffness in disordered systems-an exactly soluble model , 1974 .

[12]  K. Binder,et al.  Theory for the dynamics of clusters near the critical point. I. Relaxation of the Glauber kinetic Ising model , 1975 .

[13]  A. Coniglio.  Some cluster-size and percolation problems for interacting spins , 1976 .

[14]  D. Gaunt,et al.  The percolation probability for the site problem on the face-centred cubic lattice , 1976 .

[15]  Michael E. Fisher,et al.  The theory of condensation and the critical point , 1967 .

[16]  E. Stoll,et al.  Cluster shapes in lattice gases and percolation , 1975 .

[17]  D. S. Gaunt,et al.  A note on the mean size of clusters in the Ising model , 1976 .

[18]  J. Lebowitz,et al.  Random Spin Systems: Some Rigorous Results , 1968 .

[19]  H. Müller-Krumbhaar,et al.  The droplet model in three dimensions: Monte Carlo calculation results , 1974 .

[20]  C. Fortuin,et al.  On the random-cluster model: I. Introduction and relation to other models , 1972 .

[21]  U. Krey.  On the critical behaviour of disordered systems , 1975 .

[22]  C. Domb,et al.  Configurational studies of the Potts models , 1974 .

[23]  D. Stauffer,et al.  Possible symmetrization of Fisher's droplet picture nearTc: A bubble-droplet formula , 1971 .

[24]  Percolation theory and the Ising model , 1973 .

[25]  R. Kikuchi,et al.  Remarks on Magnetically Dilute Systems , 1959 .

[26]  B. Heap,et al.  Theory of random dilute magnets with application to MnZnF2 , 1962, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[27]  S. Kirkpatrick.  Percolation and Conduction , 1973 .

[28]  M. Fisher.  The theory of equilibrium critical phenomena , 1967 .

[29]  D. Amit.  Renormalization of the Potts model , 1976 .

[30]  H. Müller-krumbhaar.  Percolation in a lattice system with particle interaction , 1974 .

[31]  D. Rapaport.  The Ising ferromagnet with impurities : a series expansion approach: I , 1972 .

[32]  T. E. Harris.  A lower bound for the critical probability in a certain percolation process , 1960, Mathematical Proceedings of the Cambridge Philosophical Society.

[33]  Scott Kirkpatrick,et al.  An introduction to percolation theory , 1971 .

[34]  T. Odagaki,et al.  Percolation Approach to the Metal-Insulator Transition in Super-Critical Fluid Metals , 1975 .

[35]  T. Lubensky,et al.  Renormalization-Group Approach to Percolation Problems , 1975 .

[36]  T. Lubensky,et al.  Critical properties of two tensor models with application to the percolation problem , 1976 .

[37]  K. Binder,et al.  Investigation of metastable states and nucleation in the kinetic Ising model , 1974 .

[38]  D. Gaunt,et al.  Percolation processes in two dimensions. II. Critical concentrations and the mean size index , 1976 .

[39]  D. Stauffer.  Cluster structure for dilute ferromagnets and percolation , 1975 .

[40]  Tom C. Lubensky,et al.  Critical Properties of Spin-Glasses , 1976 .

[41]  D. Stauffer.  Crossover scaling, correlation function and connectivity in dilute low temperature ferromagnets , 1975 .

[42]  R. Kikuchi.  Concept of the Long-Range Order in Percolation Problems , 1970 .

[43]  K. Binder,et al.  Evidence for Fisher's Droplet Model in Simulated Two-Dimensional Cluster Distributions , 1972 .

[44]  A. Young.  The critical behaviour of disordered magnetic systems-an exactly solvable model , 1976 .

[45]  A remark to Harris's theorem on percolation , 1975 .

[46]  Michael E. Fisher,et al.  Critical Probabilities for Cluster Size and Percolation Problems , 1961 .