Communications in Mathematical Physics

, Volume 99, Issue 2, pp 253–269 | Cite as

Thermodynamic inequalities for percolation

  • R. Durrett
  • B. Nguyen


In this paper we describe the percolation analogues of the Gibbs and Helmholtz potentials and use these quantities to prove some general inequalities concerning the critical exponents of percolation processes.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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  1. 1.
    Aizenman, M.: Geometric analysis ofφ 4 fields and Ising models. Parts I and II. Commun. Math. Phys.86, 1–48 (1982)Google Scholar
  2. 2.
    Aizenman, M., Newman, C.N.: Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys. (to appear 1984)Google Scholar
  3. 3.
    Amit, D.J.: Field theory, the renormalization group, and critical phenomena. New York: McGraw-Hill 1978Google Scholar
  4. 4.
    Athreya, K., Ney, P.: Branching processes. Berlin, Heidelberg, New York: Springer 1972Google Scholar
  5. 5.
    Dhar, D., Barna, M.: Monte Carlo simulation of directed percolation on a square lattice. J. Phys. C.14, L1-L6 (1981)Google Scholar
  6. 6.
    Domany, E., Kinzel, W.: Directed percolation in two dimensions: numerical analysis and an exact solution. Phys. Rev. Lett.47, 5–8 (1981)Google Scholar
  7. 7.
    Durrett, R.: Conditioned limit theorems for some null recurrent Markov processes. Ann. Prob.6, 798–828 (1978)Google Scholar
  8. 8.
    Durrett, R.: Oriented percolation in two dimensions. Ann. Prob.12, 999–1040 (1984)Google Scholar
  9. 9.
    Durrett, R.: Some general results concerning the critical exponents of percolation processes. ZFW (to appear 1985)Google Scholar
  10. 10.
    Essam, J.W., Gwilym, K.M.: The scaling laws for percolation processes. J. Phys. C4, L228–232 (1971)Google Scholar
  11. 11.
    Essam, J.W.: Percolation theory. Rep. Prog. Phys.43, 833–911 (1980)Google Scholar
  12. 12.
    Feller, W.: Theory of probability and its applications, Vol. II. New York: Wiley 1970Google Scholar
  13. 13.
    Fisher, M.E., Essam, J.W.: Some cluster size and percolation problems. J. Math. Phys.2, 609–619 (1961)Google Scholar
  14. 14.
    Fisher, M.E.: The theory of equilibrium critical phenomena. Rep. Prog. Phys.30, 615–730 (1967)Google Scholar
  15. 15.
    Griffiths, R.B.: Ferromagnets and simple fluids near the critical point: Some thermodynamic inequalities. J. Chem. Phys.43, 1958–1968 (1965)Google Scholar
  16. 16.
    Grimmett, G.R.: On the differentiability of the number of clusters per vertex in the percolation model. J. London Math. Soc.23, 372–384 (1981)Google Scholar
  17. 17.
    Kesten, H.: Analyticity properties and power law estimates of functions in percolation theory. J. Stat. Phys.25, 717–756 (1981)Google Scholar
  18. 18.
    Kesten, H.: Percolation theory for mathematicians. Boston: Birkhäuser 1982Google Scholar
  19. 19.
    Kunz, H., Soulliard, B.: Essential singularity in percolation problems and asymptotic behavior of cluster size distributions. J. Stat. Phys.19, 77–106 (1978)Google Scholar
  20. 20.
    Nienhuis, B., Riedel, E.K., Schick, M.: Magnetic exponents of the two dimensionalq-state Potts model. J. Phys. A13, L189-L192 (1980)Google Scholar
  21. 21.
    Rushbrooke, G.S.: On the thermodynamics of the critical region for the Ising problem. J. Chem. Phys.39, 842–843 (1963)Google Scholar
  22. 22.
    Sokal, A.: More inequalities for critical exponents. J. Stat. Phys.25, 25–51 (1981)Google Scholar
  23. 23.
    Stanley, H.E.: Introduction to phase transitions and critical phenomena. Oxford: Oxford University Press 1971Google Scholar
  24. 24.
    Stoer, J., Witzgall, C.: Convexity and optimization in finite dimensions. Berlin, Heidelberg, New York: Springer 1970Google Scholar
  25. 25.
    Wu, F.Y.: Percolation and the Potts model. J. Stat. Phys.18, 115–123 (1978)Google Scholar
  26. 26.
    Wu, F.Y.: Domany-Kinzel model of directed percolation: formulation as a random walk problem and some exact results. Phys. Rev. Lett.48, 775–778 (1982)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • R. Durrett
    • 1
  • B. Nguyen
    • 1
  1. 1.Department of MathematicsU.C.L.A.Los AngelesUSA

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