Random walks in random environments

Journal of Physics A: Mathematical and General

Volumn 39, Issue 40, 2006, Pages

  Zeitouni, Ofer ^a,b  

^a University of Minnesota   (United States)

^b TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33748959874     PISSN: 03054470     EISSN: 13616447     Source Type: Journal    
DOI: 10.1088/0305-4470/39/40/R01     Document Type: Article

References (109)

1. Alili S 1999 Asymptotic behaviour for random walks in random environments J. Appl. Prob. 36 334-49
2. Anshelevich V V, Khanin K M and Sinai Ya G 1982 Symmetric random walks in random environments Commun. Math. Phys. 85 449-70
3. Bandyopadhyay A and Zeitouni O 2006 Random walk in dynamic Markovian random environment ALEA 1 205-24
4. Berger N 2006 On the limiting velocity of high-dimensional random walk in random environment Preprint math.PR/0601656
5. Berger N, Gantert N and Peres Y 2003 The speed of biased random walk on percolation clusters Probab. Theory Relat. Fields 126 221-42
6. Berger N and Biskup M 2005 Quenched invariance principles for simple random walk on percolation clusters Preprint math.PR/0503576 (Probab. Theory Relat. Fields at press)
7. Boldrighini C, Minlos R A and Pellegrinotti A 1997 Almost-sure central limit theorem for a Markov model of random walk in dynamical random environment Probab. Theory Relat. Fields 109 245-73
8. Boldrighini C, Minlos R A and Pellegrinotti A 2000 Random walk in a fluctuating random environment with Markov evolution On Dobrushin's Way: From Probability Theory to Statistical Physics (American Mathematical Society Transl. Ser. 2 vol 198) (Providence, RI: American Mathematical Society) pp 13-5
9. Boldrighini C, Minlos R A and Pellegrinotti A 2004 Random walks in quenched i.i.d. space-time random environment are always a.s. diffusive Probab. Theory Relat. Fields 129 133-56
10. Bolthausen E and Goldsheid I 2000 Recurrence and transience of random walks in random environments on a strip Commun. Math. Phys. 214 429-47
11. Bolthausen E and Sznitman A S 2002 On the static and dynamic points of view for certain random walks in random environment Methods Appl. Anal. 9 345-75
12. Bolthausen E and Zeitouni O 2006 Multiscale analysis of exit distributions for random walks in random environments Preprint math.PR/0607192
13. Bolthausen E, Sznitman A S and Zeitouni O 2003 Cut points and diffusive random walks in random environments Ann. Inst. Henri Poincare 39 527-55
14. Bouchaud J P, Georges A and Le Doussal P 1987 Anomalous diffusion in random media: trapping, correlations and central limit theorems J. Physique 48 1855-60
15. Boivin D and Depauw J 2003 Spectral homogenization of reversible random walks on in a random environment Stoch. Proc. Appl. 104 29-56
16. Bramson M 1991 Random walk in random environment: a counterexample without potential J. Stat. Phys. 62 863-75
17. Bramson M and Durrett R 1988 Random walk in random environment: a counterexample? Commun. Math. Phys. 119 199-211
18. Bramson M, Zeitouni O and Zerner M P W 2006 Shortest spanning trees and a counterexample for random walks in random environments Ann. Probab. 34 821-56
19. Brémont J 2004 Behavior of random walks on in Gibbsian medium C. R. Math. Acad. Sci. Paris 338 895-8
20. Brémont J 2004 Julien Random walks in random medium on and Lyapunov spectrum Ann. Inst. Henri Poincare Probab. Stat. 40 309-36
21. Bricmont J and Kupiainen A 1991 Random walks in asymmetric random environments Commun. Math. Phys 142 345-420
22. Brox T 1986 A one-dimensional diffusion process in a Wiener medium Ann. Probab. 14 1206-18
23. Cheliotis D 2005 Diffusions in random environments and the renewal theorem Ann. Probab. 33 1760-81
24. Comets F, Gantert N and Zeitouni O 2000 Quenched, annealed and functional large deviations for one dimensional random walk in random environment Probab. Theory Relat. Fields 118 65-114
25. Comets F and Zeitouni O 2004 A law of large numbers for random walks in random mixing environments Ann. Probab. 32 880-914
26. Comets F and Zeitouni O 2005 Gaussian fluctuations for random walks in random mixing environments Isr. J. Math. 148 87-114
27. Dembo A, Gantert N, Peres Y and Shi Z 2005 Valleys and the maximum local time for random walk in random environment Preprint math.PR/0508579
28. Dembo A and Zeitouni O 1998 Large Deviations Techniques and Applications 2nd edn (Berlin: Springer)
29. Dembo A, Guionnet A and Zeitouni O 2001 Aging properties of Sinai's random walk in random environment Preprint math.PR/0105215
30. Dembo A, Gantert N, Peres Y and Zeitouni O 2002 Large deviations for random walks on Galton-Watson trees: averaging and uncertainty Probab. Theory Relat. Fields 122 241-88
31. Dembo A, Gantert N and Zeitouni O 2004 Large deviations for random walk in random environment with holding times Ann. Probab. 32 996-1029
32. Dembo A, Peres Y and Zeitouni O 1996 Tail estimates for one-dimensional random walk in random environment Commun. Math. Phys. 181 667-84
33. Derriennic Y and Lin M 2003 The central limit theorem for Markov chains started at a point Probab. Theory Relat. Fields 125 73-6
34. Derrida B and Luck J M 1983 Diffusion on a random lattice: weak-disorder expansion in arbitrary dimension Phys. Rev. B 28 7183-90
35. Donsker M D and Varadhan S R S 1983 Asymptotic evaluation of certain Markov process expectations for large time, IV Commun. Pure Appl. Math. 36 183-212
36. Doyle P G and Snell J L 1984 Random Walks and Electric Networks (Carus Mathematical Monographs vol 22) (Washington, DC: Mathematical Association of America)
37. Erdös P and Taylor S J 1960 Some intersection properties of random walks paths Acta Math. Acad. Sci. Hungar. 11 231-48
38. Fisher D S 1984 Random walks in random environments Phys. Rev. A 30 960-4
39. Gantert N 2002 Subexponential tail asymptotics for a random walk with randomly placed one-way nodes Ann. Inst. Henri Poincare 38 1-16
40. Gantert N and Shi Z 2002 Many visits to a single site by a transient random walk in random environment Stoch. Process. Appl. 99 159-76
41. Gantert N and Zeitouni O 1998 Quenched sub-exponential tail estimates for one-dimensional random walk in random environment Commun. Math. Phys. 194 177-90
42. Goergen L 2006 Limit velocity and zero-one laws for diffusions in random environment Preprint math.PR/0512061 (Ann. Appl. Probab. at press)
43. Goldsheid I 2006 Simple transient random walks in one-dimensional random environment: the central limit theorem Preprint math.PR/0605775
44. Golosov A O 1985 On limiting distributions for a random walk in a critical one dimensional random environment Commun. Moscow Math. Soc. 199 199-200
45. Greven A and den Hollander F 1994 Large deviations for a random walk in random environment Ann. Probab. 22 1381-428
46. Hu Y and Shi Z 2000 The problem of the most visited site in random environment Probab. Theory Relat. Fields 116 273-302
47. Hu Y and Shi Z 2006 A subdiffusive behavior of recurrent random walk in random environment on a regular tree Preprint math.PR/0603363
48. Hughes B D 1996 Random Walks and Random Environments (Oxford: Oxford University Press)
49. Kalikow S A 1981 Generalized random walks in random environment Ann. Probab. 9 753-68
50. Kesten H 1986 The limit distribution of Sinai's random walk in random environment Physica A 138 299-309
51. Key E S 1984 Recurrence and transience criteria for random walk in a random environment Ann. Probab. 12 529-60
52. Kosygina E, Rezakhanlou F and Varadhan S R S 2006 Stochastic homogenization of Hamilton-Jacobi-Bellman equations Commun. Pure Appl. Math. at press
53. Kesten H, Kozlov M V and Spitzer F 1975 A limit law for random walk in a random environment Comput. Math. 30 145-68
54. Kozlov S M 1985 The method of averaging and walks in inhomogeneous environments Russ. Math. Surv. 40 73-145
55. Kunnemann R 1983 The diffusion limit of reversible jump processes in with ergodic random bond conductivities Commun. Math. Phys. 90 27-68
56. Kuo H-J and Trudinger N S 1990 Linear elliptic difference inequalities with random coefficients Math. Comput. 55 37-53
57. Lawler G F 1982 Weak convergence of a random walk in a random environment Commun. Math. Phys. 87 81-7
58. Lawler G F 1989 Low-density expansion for a two-state random walk in a random environment J. Math. Phys. 30 145-57
59. Lawler G F 1991 Intersections of Random Walks (Basle: Birkhauser)
60. Le Doussal P, Monthus C and Fisher D 1999 Random walkers in one-dimensional random environment: exact renormalization group analysis Phys. Rev. E 59 4795-4840
61. Ledrappier F 1984 Quelques Propriétés Des Exposants Charactéristiques (Lecture Notes in Mathematics vol 1097) (New York: Springer)
62. Liggett T M 1985 An improved subadditive ergodic theorem Ann. Probab. 13 1279-85
63. Lyons R and with Peres Y Probability on trees and networks Available at http://mypage.iu.edu/∼rdlyons/prbtree/prbtree.html
64. De Masi A, Ferrari P A, Goldstein S and Wick W D 1989 An invariance principle for reversible Markov processes: applications to random motions in random environments J. Stat. Phys. 55 787-855
65. Mayer-Wolf E, Roitershtein A and Zeitouni O 2004 Limit theorems for one-dimensional transient random walks in Markov environments Ann. Inst. Henri Poincare 40 635-59 (CE: Check it)
66. Mathieu P 1984 Zero white noise limit through Dirichlet forms, with applications to diffusions in a random medium Probab. Theory Relat. Fields 99 549-80
67. Mathieu P 1985 Limit theorems for diffusions with a random potential Stoch. Proc. Appl. 60 103-11
68. Mathieu P and Piatnitski A 2005 Quenched invariance principles for random walks on percolation clusters Preprint math.PR/0505672
69. Molchanov S A 1994 Lectures on random media (Lecture Notes in Mathematics vol 1581) (New York: Springer)
70. Papanicolaou G C and Varadhan S R S 1982 Diffusions with random coefficients Statistics and Probability: Essays in Honor of C. R. Rao (Amsterdam: North-Holland) pp 547-52
71. Peres Y and Zeitouni O 2006 A central limit theorem for biased random walks on Galton Watson trees Preprint math.PR/0606625
72. Peterson J 2007 PhD Thesis University of Minnesota
73. Pisztora A and Povel T 1999 Large deviation principle for random walk in a quenched random environment in the low speed regime Ann. Probab. 27 1389-413
74. Pisztora A, Povel T and Zeitouni O 1999 Precise large deviation estimates for a one-dimensional random walk in a random environment Probab. Theory Relat. Fields 113 191-219
75. Rassoul-Agha F 2003 The point of view of the particle on the law of large numbers for random walks in a mixing random environment Ann. Probab. 31 1441-63
76. Rassoul-Agha F 2004 Large deviations for random walks in a mixing random environment and other (non-markov) random walks Commun. Pure Appl. Math 57 1178-96
77. Rassoul-Agha F and Seppäläinen T 2005 An almost sure invariance principle for random walks in a space-time random environment Probab. Theory Relat. Fields 133 299-314
78. Révész P 2005 Random Walk in Random and Non-Random Environments 2nd edn (Singapore: World Scientific)
79. Revuz D and Yor M 1999 Continuous Martingales and Brownian Motion 3rd edn (Berlin: Springer)
80. Sabot C 2004 Random walks in random environment at low disorder Ann. Probab. 32 2996-3023
81. Schmitz T 2005 Diffusions in random environment and ballistic behavior Preprint math.PR/0509554
82. Schmitz T 2006 Examples of condition (T) for diffusions in random environment Preprint math.PR/0607293
83. Schumacher S 1985 Diffusions with random coefficients Contemp. Math. 41 351-6
84. Shen L 2002 Asymptotic properties of certain anisotropic walks in random media Ann. Appl. Probab. 12 477-510
85. Shen L 2003 On ballistic diffusions in random environment Ann. Inst. Henri Poincare 39 839-76
86. Shi Z 2001 Sinai's walk via stochastic calculus Milieux Aléatoires (Panoramas et Synthèses vol 12) ed F Comets and E Pardoux ((Societe Mathematique de France)) pp 53-74
87. Sidoravicius V and Sznitman A S 2004 Quenched invariance principles for walks on clusters of percolation or among random conductances Probab. Theory Related Fields 129 219-44
88. Sinai Ya G 1982 The limiting behavior of a one-dimensional random walk in random environment Theor. Prob. Appl. 27 256-68
89. Solomon F 1975 Random walks in random environments Ann. Probab. 3 1-31
90. Spitzer F 1976 Principles of Random Walk 2nd edn (Berlin: Springer)
91. Stroock D and Varadhan S R S 1979 Multidimensional Diffusion Processes (Berlin: Springer)
92. Stannat W 2004 A remark on the CLT for a random walk in a random environment Probab. Theory Related Fields 130 377-87
93. Sznitman A S 1998 Brownian Motion, Obstacles and Random Media (Berlin: Springer)
94. Sznitman A S 1999 On a class of transient random walks in random environment Ann. Probab. 29 724-65
95. Sznitman A S 2000 Slowdown estimates and central limit theorem for random walks in random environment J. Emergency Med. Serv. 2 93-143
96. Sznitman A S 2002 An effective criterion for ballistic behavior of random walks in random environment Probab. Theory Relat. Fields 122 509-44
97. Sznitman A S 2003 On new examples of ballistic random walks in random environment Ann. Probab. 31 285-322
98. sznitman A S 2003 On the anisotropic walk on the supercritical percolation cluster Commun. Math. Phys. 240 123-48
99. Sznitman A S 2004 Topics in random walks in random environment School and Conference on Probability Theory (ICTP Lect. Notes vol XVII) (Trieste: Abdus Salam Int. Cent. Theoret. Phys.) pp 203-266
100. Sznitman A S and Zerner M 1999 A law of large numbers for random walks in random environment Ann. Probab. 27 1851-69
101. Sznitman A S and Zeitouni O 2006 An invariance principle for isotropic diffusions in random environments Invent. Math. 164 455-567
102. Temkin D E 1972 One dimensional random walk in a two-component chain Sov. Math. Dokl. 13 1172-6
103. Varadhan S R S 1966 Asymptotic probabilities and differential equations Commun. Pure Appl. Math. 19 261-86
104. Varadhan S R S 2003 Large deviations for random walks in a random environment Commun. Pure Appl. Math. 56 1222-45
105. Zeitouni O 2002 Random walks in Random Environments: Proc. ICM 2002 (Documenta Mathematica vol III) (Beijing: Higher Ed press) pp 117-27
106. Zeitouni O 2004 Random walks in random environment XXXI Summer School in Probability (St. Flour, 2001) (Lecture Notes in Mathematics vol 1837) (Berlin: Springer) pp 193-312
107. Zerner M P W 1998 Lyapounov exponents and quenched large deviations for multidimensional random walk in random environment Ann. Probab. 26 1446-76
108. Zerner M P W 2002 A non-ballistic law of large numbers for random walks in i.i.d. random environment Electron. Commun. Probab. 7 191-7 (paper 19)
109. Zerner M P W and Merkl F 2001 A zero-one law for planar random walks in random environment Ann. Probab. 29 1716-32