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On the upper critical dimension of lattice trees and lattice animals

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Abstract

We give a rigorous proof of mean-field critical behavior for the susceptibility (γ=1/2) and the correlation length (v=1/4) for models of lattice trees and lattice animals in two cases: (i) for the usual model with trees or animals constructed from nearest-neighbor bonds, in sufficiently high dimensions, and (ii) for a class of “spread-out” or long-range models in which trees and animals are constructed from bonds of various lengths, above eight dimensions. This provides further evidence that for these models the upper critical dimension is equal to eight. The proof involves obtaining an infrared bound and showing that a certain “square diagram” is finite at the critical point, and uses an expansion related to the lace expansion for the self-avoiding walk.

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References

  1. D. C. Brydges and T. Spencer, Self-avoiding walk in 5 or more dimensions,Commun. Math. Phys. 97:125–148 (1985).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. G. Slade, The diffusion of self-avoiding random walk in high dimensions,Commun. Math. Phys. 110:661–683 (1987).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. G. Slade, The scaling limit of self-avoiding random walk in high dimensions,Ann. Prob. 17:91–107 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  4. G. Slade, Convergence of self-avoiding random walk to Brownian motion in high dimensions,J. Phys. A 21:L417-L420 (1988).

    Article  MathSciNet  ADS  Google Scholar 

  5. G. Lawler, The infinite self-avoiding walk in high dimensions,Ann. Prob. 17:1367–1376 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  6. T. Hara and G. Slade, Mean-field critical behaviour for percolation in high dimensions,Commun. Math. Phys. 128:333–391 (1990); The triangle condition for percolation,Bull. AMS (New Series) 21:269–273 (1989).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. M. Aizenman and C. M. Newman, Tree graph inequalities and critical behaviour in percolation models,J. Stat. Phys. 36:107–143 (1984).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. D. J. Barsky and M. Aizenman, Percolation critical exponents under the triangle condition, Preprint (1988).

  9. B. G. Nguyen, Gap exponents for percolation processes with triangle condition,J. Stat. Phys. 49:235–243 (1987).

    Article  ADS  MATH  Google Scholar 

  10. T. Hara, Mean field critical behaviour for correlation length for percolation in high dimensions,Prob. Theory Rel. Fields, in press.

  11. J. Fröhlich, B. Simon, and T. Spencer, Infrared bounds, phase transitions, and continuous symmetry breaking,Commun. Math. Phys. 50:79–95 (1976).

    Article  ADS  Google Scholar 

  12. M. Aizenman, Geometric analysis ofϕ 4 fields and Ising models, Parts I and II,Commun. Math. Phys. 86:1–48 (1982).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. J. Fröhlich, On the triviality ofϕ 4 d theories and the approach to the critical point ind⩾4 dimensions,Nucl. Phys. B 200[FS4]:281–296 (1982).

    Article  ADS  Google Scholar 

  14. J. W. Essam, Percolation theory,Rep. Prog. Phys. 43:833–912 (1980).

    Article  MathSciNet  ADS  Google Scholar 

  15. T. C. Lubensky and J. Isaacson, Statistics of lattice animals and dilute branched polymers,Phys. Rev. A 20:2130–2146 (1979).

    Article  ADS  Google Scholar 

  16. A. Bovier, J. Fröhlich, and U. Glaus, Branched polymers and dimensional reduction, inCritical Phenomena, Random Systems, Gauge Theories, K. Osterwalder and R. Stora, eds. (North-Holland, Amsterdam, 1984).

    Google Scholar 

  17. H. Tasaki, Stochastic geometric methods in statistical physics and field theories, Ph.D. thesis, University of Tokyo (1986).

  18. H. Tasaki and T. Hara, Critical behaviour in a system of branched polymers,Prog. Theor. Phys. Suppl. 92:14–25 (1987).

    Article  MathSciNet  ADS  Google Scholar 

  19. J. T. Chayes and L. Chayes, On the upper critical dimension of Bernoulli percolation,Commun. Math. Phys. 113:27–48 (1987).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. H. Tasaki, Hyperscaling inequalities for percolation,Commun. Math. Phys. 113:49–65 (1987).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. H. Tasaki, Geometric critical exponent inequalities for general random cluster models,J Stat. Phys. 49:841–847 (1987).

    Article  MathSciNet  ADS  Google Scholar 

  22. D. J. Klein, Rigorous results for branched polymer models with excluded volume,J. Chem. Phys. 75:5186–5189 (1981).

    Article  ADS  Google Scholar 

  23. D. A. Klarner, Cell growth problems,Can. J. Math. 19:851–863 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  24. N. Madras, C. E. Soteros, and S. G. Whittington, Statistics of lattice animals,J. Phys. A 21:4617–4635 (1988).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. G. Parisi and N. Sourlas, Critical behaviour of branched polymers and the Lee-Yang edge singularity,Phys. Rev. Lett. 46:871–874 (1981).

    Article  MathSciNet  ADS  Google Scholar 

  26. M. V. Menshikov, S. A. Molchanov, and A. F. Sidorenko, Percolation theory and some applications,Itogi Nauki Tekhniki (Ser. Prob. Theory, Math. Stat., Theoret. Cybernet.) 24:53–110 (1986) [English transl.,J. Sov. Math. 42:1766–1810 (1988)].

    MathSciNet  Google Scholar 

  27. J. van den Berg and H. Kesten, Inequalities with applications to percolation and reliability,J. Appl. Prob. 22:556–569 (1985).

    Article  MATH  Google Scholar 

  28. C. Berge,Graphs and Hypergraphs (North-Holland, Amsterdam, 1973).

    MATH  Google Scholar 

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Hara, T., Slade, G. On the upper critical dimension of lattice trees and lattice animals. J Stat Phys 59, 1469–1510 (1990). https://doi.org/10.1007/BF01334760

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