Skip to main content
Log in

Thermodynamic inequalities for percolation

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  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. Aizenman, M., Newman, C.N.: Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys. (to appear 1984)

  3. Amit, D.J.: Field theory, the renormalization group, and critical phenomena. New York: McGraw-Hill 1978

    Google Scholar 

  4. Athreya, K., Ney, P.: Branching processes. Berlin, Heidelberg, New York: Springer 1972

    Google Scholar 

  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. 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. Durrett, R.: Conditioned limit theorems for some null recurrent Markov processes. Ann. Prob.6, 798–828 (1978)

    Google Scholar 

  8. Durrett, R.: Oriented percolation in two dimensions. Ann. Prob.12, 999–1040 (1984)

    Google Scholar 

  9. Durrett, R.: Some general results concerning the critical exponents of percolation processes. ZFW (to appear 1985)

  10. Essam, J.W., Gwilym, K.M.: The scaling laws for percolation processes. J. Phys. C4, L228–232 (1971)

    Google Scholar 

  11. Essam, J.W.: Percolation theory. Rep. Prog. Phys.43, 833–911 (1980)

    Google Scholar 

  12. Feller, W.: Theory of probability and its applications, Vol. II. New York: Wiley 1970

    Google Scholar 

  13. Fisher, M.E., Essam, J.W.: Some cluster size and percolation problems. J. Math. Phys.2, 609–619 (1961)

    Google Scholar 

  14. Fisher, M.E.: The theory of equilibrium critical phenomena. Rep. Prog. Phys.30, 615–730 (1967)

    Google Scholar 

  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. 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. Kesten, H.: Analyticity properties and power law estimates of functions in percolation theory. J. Stat. Phys.25, 717–756 (1981)

    Google Scholar 

  18. Kesten, H.: Percolation theory for mathematicians. Boston: Birkhäuser 1982

    Google Scholar 

  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. 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. Rushbrooke, G.S.: On the thermodynamics of the critical region for the Ising problem. J. Chem. Phys.39, 842–843 (1963)

    Google Scholar 

  22. Sokal, A.: More inequalities for critical exponents. J. Stat. Phys.25, 25–51 (1981)

    Google Scholar 

  23. Stanley, H.E.: Introduction to phase transitions and critical phenomena. Oxford: Oxford University Press 1971

    Google Scholar 

  24. Stoer, J., Witzgall, C.: Convexity and optimization in finite dimensions. Berlin, Heidelberg, New York: Springer 1970

    Google Scholar 

  25. Wu, F.Y.: Percolation and the Potts model. J. Stat. Phys.18, 115–123 (1978)

    Google Scholar 

  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 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by J. Fröhlich

Partially supported by NSF grant MCS 83-00836, this author is an AMS Postdoctoral Research Fellow 1984–1986

Rights and permissions

Reprints and permissions

About this article

Cite this article

Durrett, R., Nguyen, B. Thermodynamic inequalities for percolation. Commun.Math. Phys. 99, 253–269 (1985). https://doi.org/10.1007/BF01212282

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01212282

Keywords

Navigation