Subdiffusive behavior of random walk on a random cluster

 Access to full text

 Full (PDF)

1. [1] S. Alexander and R. Orbach, Density of states on fractals: « fractons » ; J. Physique Lett., t. 43, 1982, L625-631. 
2. [2] K.B. Athreya and P.E. Ney, Branching Processes, 1972, Springer-Verlag. Zbl0259.60002MR373040
3. [3] Y.S. Chow and H. Teicher, Probability Theory, 1978, Springer-Verlag. Zbl0399.60001MR513230
4. [4] P.G. De Gennes, La percolation : un concept unificateur, La Recherche, t. 7, 1976, p. 919-927. 
5. [5] A. De Masi, P.A. Ferrari, S. Goldstein and W.D. Wick, An invariance principle for reversible Markov processes with application to random motions in random environments, (1985, preprint). Zbl0713.60041
6. [6] P.A. Doyle and J.L. Snell, Random Walk and Electric Networks, Carus Math. Monograph, No. 22, 1984, Math. Assoc. of America. Zbl0583.60065MR920811
7. [7] R. Durrett, Conditioned limit theorems for some null recurrent Markov processes, Ann. Probab., t. 6, 1978, p. 798-828. Zbl0398.60023MR503953
8. [8] W. Feller, An Introduction to Probability Theory and its Applications, Vol. I, 3rd ed., John Wiley and Sons, 1968. Zbl0155.23101MR228020
9. [9] D. Griffeath and T.M. Liggett, Critical phenomena for Spitzer's reversible nearest particle systems, Ann. Probab., t. 10, 1982, p. 881-895. Zbl0498.60090MR672290
10. [10] T.E. Harris, The Theory of Branching Processes, Springer-Verlag and Prentice Hall, 1963. Zbl0117.13002MR163361
11. [11] S. Havlin, D. Movshovitz, B. Trus and G.H. Weiss, Probability densities for the displacement of random walks on percolation clusters, J. Phys. A. Math. Gen., t. 18, 1985, L719-722. 
12. [12] C.C. Heyde, On large deviation probabilities in the case of attraction to a non-normal stable law, Sankhya, Ser. A, t. 30, 1968, p. 253-258. Zbl0182.22903MR240854
13. [13] P. Jagers, Branching Processes with Biological Applications, John Wiley and Sons, 1950. Zbl0356.60039
14. [14] H. Kesten, The critical probability of bond percolation on the square lattice equals 1/2, Comm. Math. Phys., t. 74, 1980, p. 41-59. Zbl0441.60010MR575895
15. [15] H. Kesten, Percolation Theory for Mathematicians, Birkhauser-Boston, 1982. Zbl0522.60097MR692943
16. [16] H. Kesten, The incipient infinite cluster in two-dimensional percolation, to appear in Theor. Prob. Rel. Fields, 1986. Zbl0584.60098MR859839
17. [17] R. Künnemann, The diffusion limit for reversible jump processes on Zd with periodic random bond conductivities, Comm. Math. Phys., t. 90, 1983, p. 27-68. Zbl0523.60097MR714611
18. [18] F. Leyvraz and H.E. Stanley, To what class of fractals does the Alexander-Orbach conjecture apply?Phys. Rev. Lett., t. 51, 1983, p. 2048-2051. 
19. [19] M. Loeve, Probability Theory, 4th ed., Springer Verlag, 1977. Zbl0359.60001MR651017
20. [20] B.B. Mandelbrot and S. Kirkpatrick, Solvable fractal family, and its possible relation to the backbone at percolation, Phys. Rev. Lett., t. 47, 1981, p. 1771-1774. MR637547
21. [21] P.A. Meyer, Martingales and Stochastic Integrals I, Lecture Notes in Math, t. 284, 1972, Springer-Verlag. Zbl0239.60001MR426145
22. [22] C.D. Mitescu and J. Roussenq, Diffusion on percolation structures, in Percolation Structures and Processes, Ann. Israel. Phys. Soc., t. 5, 1983, Eds. G. Deutscher, R. Zallen and J. Adler. 
23. [23] A.G. Pakes, Some limit theorems for the total progeny of a branching process, Adv. Appl. Prob., t. 3, 1971, p. 176-192. Zbl0218.60075MR283892
24. [24] R. Rammal and G. Toulouse, Random walk on fractal structures and percolation clusters, J. Physique-Lett., t. 44, 1983, L13-22. 
25. [25] P.D. Seymour and D.J.A. Welsh, Percolation probabilities on the square lattice, Ann. Discrete Math., t. 3, 1978, p. 227-245. Zbl0405.60015MR494572
26. [26] R.S. Slack, A branching process with mean one and possibly infinite variance, Z. Wahrsch. verw. Geb., t. 9, 1968, p. 139-145. Zbl0164.47002MR228077
27. [27] J. Straley, The ant in the labyrinth: diffusion in random networks near the percolation threshold, J. Phys., Solid State Phys., t. C13, 1980, p. 2991-3002. 
28. [28] J. Van Den Berg and H. Kesten, Inequalities with applications to percolation and reliability, J. Appl. Prob., t. 22, 1985, p. 556-569. Zbl0571.60019MR799280

1. David Croydon, Convergence of simple random walks on random discrete trees to brownian motion on the continuum random tree
2. Martin T. Barlow, Richard F. Bass, The construction of brownian motion on the Sierpinski carpet
3. Martin T. Barlow, Yuval Peres, Perla Sousi, Collisions of random walks
4. David Aldous, Jim Pitman, Tree-valued Markov chains derived from Galton-Watson processes
5. Romain Abraham, Jean-François Delmas, Hui He, Pruning Galton–Watson trees and tree-valued Markov processes