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Markov Cascades

  • Edward C. Waymire
  • Stanley C. Williams
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 84)

Abstract

The Kahane-Peyrière theory of homogeneous independent cascades is extended to a class of finite state Markov dependent cascades. The problems of (i) non-triviality (ii) divergence of moments and (iii) carrying dimension are completely solved for this class of models. While the authors have obtained complete solutions to problems (i) and (iii), and partial solutions to (ii) in wider generality, the focus of the present paper is on fixed points of a naturally associated random recursive equation of independent interest in diverse contexts. By essentially algebraic methods of this paper we obtain complete solutions to all three problems (i)-(iii). In particular an interesting role of a “mean-reversal” symmetry in the computation of critical parameters for survival and carrying dimension is uncovered.

Keywords

Spin Glass Hausdorff Dimension Multifractal Spectrum Dimension Spectrum Survival Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Aizenman, M., Lebowitz, J.L., and D. Ruelle (1987): Some rigorous results on the Sherrington-Kirkpatrick spin glass model, Commun. Math. Phys. 112, 3–20.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Bhattacharya, R.N. and E. Waymire (1990): Stochastic Processes with Applications, John Wiley and Sons, New York.Google Scholar
  3. [3]
    Buffet, E., A. Patrick, J.V. Pule (1993): Directed polymers on trees: a martingale approach, J. Phys. A 26, 1823–1834.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Chayes, J., Chayes, and L. Durrett, R. (1988): Connectivity properties of Mandelbrot’s percolation, Probab. Th. Rel. Fields 77, 307–324.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Chauvin, B. and A. Rouault (1995): Boltzmann-Gibbs weights in the branching random walk, (this volume).Google Scholar
  6. [6]
    Collet,P. and F. Koukiou (1992): Large deviations for multiplicative chaos, Commun. Math. Phys. 147, 329–342.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Comets, F. and J. Neveu (1995): The Sherrington-Kirkpatrick model of spin glasses and stochastic calculus, Comm. Math. Phys. 166(3), 549–564.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Cutler, C. (1986): The Hausdorff dimension distribution of finite measures in Euclidean space, Can. J. Math. XXXVIII, 1459–1484.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Dekking, F.M., and Grimmett G.R.(1988): Superbranching processes and projections of random Cantor sets, Probab. Th. Rel. Fields 78, 335–355.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Dekking, F.M. and R. W. J. Meester (1990) On the structure of Mandelbrot’s percolation process and other random Cantor sets, Jour. of Stat. Phys. 58(5/6), 1109–1126.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Dembo, A. and O. Zeitouni (1993): Large deviation techniques and applications, Jones-Bartlett, Boston.Google Scholar
  12. [12]
    Deuschel, J-D, D. Stroock (1989): Large Deviations, Academic Press, NY.zbMATHGoogle Scholar
  13. [13]
    Dubins, L. and D. A. Freedman (1967): Random distribution functions, in Proceedings of the 5th Berkeley Symposium in Probability and Statistics, University of California Press, Berkeley.Google Scholar
  14. [14]
    Dubrulle, B. (1994): Intermittancy in fully developed turbulence: logPoisson statistics and generalized scale invariance, Phys. Rev. Lett. 73(7), 959–962.CrossRefGoogle Scholar
  15. [15]
    Derrida, B. Spohn, H. (1988): Polymers on disordered trees, spin glasses, travelling waves, Jour. Stat. Phys. 51, 817–840.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Derrida, B. (1991): Mean field theory of directed polymers in a random medium and beyond, Physica Scripta T38, 6–12.CrossRefGoogle Scholar
  17. [17]
    Falconer, K. (1993): The multifractal spectrum of statistically self-similar measures, preprint.Google Scholar
  18. [18]
    Gupta, V.K. and E. Waymire (1993): A statistical analysis of mesoscale rainfall as a random cascade, J. Appld. Meteor.32, 251–267.CrossRefGoogle Scholar
  19. [19]
    Graf, S., Mauldin, D. and S. Williams (1987): The exact Hausdorff dimension in random recursive constructions, Memoirs. Am. Math. Soc. 71 (381).Google Scholar
  20. [20]
    Holley, R. and E. Waymire (1992), Multifractal dimensions and scaling exponents for strongly bounded random cascades, Ann. Applied Probab. 2, 819–845.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    Kahane, J.P. and J. Peyrière (1976), Sur certaines martingales de Benoit Mandelbrot, Advances in Math. 22, 131–145.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Kahane, J.P. (1985): Sur le chaos multiplicatif, Ann. Sciences Math. Quebec 9, 105–150.MathSciNetzbMATHGoogle Scholar
  23. [23]
    Kahane, J-P (1987): Multiplications aleatoires et dimensions de Hausdorff, Ann. Inst. Poincare 23, 289–296.MathSciNetGoogle Scholar
  24. [24]
    Kahane, J.P. (1991): Produits de poids aleatoires indpendendants et applications, Fractal Geometry and Analysis (ed. by J. Belair, S. Dubac), Kluwer Academic Publ. The Netherlands.Google Scholar
  25. [25]
    Kahane, J.P. and Y. Katznelson (1990): Decomposition des measures selon la dimension, Colloq. Math. vol LVII, 269–279MathSciNetGoogle Scholar
  26. [26]
    Koukiou, F. (1993): The mean-field theory of spin glass and directed polymer models in random media, preprint.Google Scholar
  27. [27]
    Lyons, R., R. Pemantle, and Y. Peres (1995): Ergodic theory on Galton-Watson trees I: Speed of random walk and dimension of harmonic measure, Erg odic Theory and Dynamical Systems 15, 593–619.MathSciNetGoogle Scholar
  28. [28]
    Mandelbrot, B. (1982): The Fractal Geometry of Nature, Freeman, S.F.zbMATHGoogle Scholar
  29. [29]
    Meneveau C. and K.R. Sreenivasan (1987), The multifractal spectrum of the dissipation field in turbulent flows, Nuclear Physics B (Proc. Suppl.) 2, 49–76.CrossRefGoogle Scholar
  30. [30]
    Moore, T. and J.L. Snell (1979): A branching process showing phase transition, J. Appld. Probab. 16, 252–260.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    Nasr, Ben (1987): Mesures aleatores de Mandelbrot asociees a des substitutions, C.R. Acad. Sc. Paris 304, 255–258.zbMATHGoogle Scholar
  32. [32]
    Ossiander, M., E. Waymire, S.C. Williams (1994): All temperature bounds on Sherrington-Kirkpatrick spin glass and T-martingales.Google Scholar
  33. [33]
    Peckham, S. and E. Waymire (1991): On a symmetry of turbulence, Comm. Math. Phys 147, 365–370.MathSciNetCrossRefGoogle Scholar
  34. [34]
    Peyrière, J.(1977): Calculs de dimensions de Hausdorff, Duke Mathematical Journals 44, 591–601.zbMATHCrossRefGoogle Scholar
  35. [35]
    Preston, C. (1974): Gibbs States on Countable Sets,Cambridge Tracts in Mathematics 68, Cambridge University Press.zbMATHCrossRefGoogle Scholar
  36. [36]
    She, Z.S. and E. Waymire (1995): Quantized energy cascade and logPoisson statistics in fully developed turbulencePhys. Rev. Lett. 74(2)262–265.CrossRefGoogle Scholar
  37. [37]
    Spitzer, F. (1975): Markov random fields on an infinite tree, Ann. Prob. 3, 387–398.MathSciNetzbMATHCrossRefGoogle Scholar
  38. [38]
    Tessier, Y., S. Lovejoy, and D. Schertzer (1993): Universal multifractals: theory and observation for rain and clouds, J. Appld. Meteor. 32, 223–250.CrossRefGoogle Scholar
  39. [39]
    Waymire, E. and S.C. Williams (1995a): Multiplicative cascades: dimension spectra and dependence, Jour. Fourier Analysis and Appl. Special issue in honor of J-P Kahane, 489–609.Google Scholar
  40. [40]
    Waymire, E. and S.C. Williams (1996): A cascade decomposition theory with applications to Markov and exchangeable cascades, Trans. AMS 348(2), 585–632.MathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    Waymire, E. and S.C. Williams (1995b): Correlated spin glasses on trees, preprint.Google Scholar
  42. [42]
    Waymire, E. and S.C. Williams (1994): A general decomposition theory for random cascades, Bull. AMS 31(2), 216–222.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Edward C. Waymire
    • 1
  • Stanley C. Williams
    • 2
  1. 1.Department of MathematicsOregon State UniversityCorvallis
  2. 2.Department of MathematicsUtah State UniversityLogan

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