Universal scaling behavior of non-equilibrium phase transitions

International Journal of Modern Physics B

Volumn 18, Issue 31-32, 2004, Pages 3977-4118

  Lübeck, Sven ^a  

^a UNIVERSITY OF DUISBURG ESSEN   (Germany)

Author keywords

Non equilibrium critical phenomena; Phase transitions; Scaling; Universality

Indexed keywords

ABSORPTION; ACCURACY; AUTOMATION; CHEMICAL MODIFICATION; CHEMOTAXONOMY; COMPUTER SIMULATION; CONJUGATION; CRYSTAL STRUCTURE; ELECTRIC FIELD; EQUILIBRIUM CONSTANT; EXAMINATION; FINITE ELEMENT ANALYSIS; MOLECULAR DYNAMICS; PARTICLE SIZE; PHASE TRANSITION; RESEARCH; REVIEW; SCALE UP; SENSITIVITY ANALYSIS; STEADY STATE; THEORY; THINKING; WORK;

EID: 14844354256     PISSN: 02179792     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0217979204027748     Document Type: Review

References (394)

1. H. E. Stanley, Scaling, universality, and renormalization: Three pillars of modern critical phenomena, Rev. Mod. Phys. 71, S358 (1999).
2. L. P. Kadanoff, Critical behavior, universality and scaling, Proceedings of the 1970 Varenna Summer School on Critical Phenomena, ed. M. S. Green (Academic Press, New York, 1971).
3. E. A. Guggenheim, The principle of corresponding states, J. Chem. Phys. 13, 253 (1945).
4. S. Milošević and H. E. Stanley, Equation of state near the critical point: II. Comparison with experiment and possible universality with respect to lattice structure and spin quantum number, Phys. Rev. B6, 1002 (1972).
5. A. Pelissetto and E. Vicari, Critical phenomena and renormalization-group theory, Phys. Rep. 368, 549 (2002).
6. H. Haken, Cooperative phenomena in systems far from thermal equilibrium and in nonphysical systems, Rev. Mod. Phys. 47, 67 (1975).
7. H. Hinrichsen, Nonequilibrium critical phenomena and phase transitions into absorbing states, Adv. Phys. 49, 815 (2000).
8. G. Ódor, Universality classes in nonequilibrium lattice systems, Rev. Mod. Phys. 76, 663 (2004).
9. W. Weidlich, Physics and social science - The approach of synergetics, Phys. Rev. 204, 1 (1991).
10. D. Helbing, Quantitative Sociodynamics - Stochastic Methods and Models of Social Interaction Processes (Kluwer, Amsterdam, 1995).
11. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North Holland, Amsterdam, 1992).
12. H. Risken, The Fokker-Planck Equation (Springer, Berlin, 1989).
13. S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys. 55, 601 (1983).
14. G. M. Schütz, Exactly Solvable Models for Many-Body Systems Far from Equilibrium in Phase Transitions and Critical Phenomena, Vol. 19, ed C. Domb and J. L. Lebowitz (Academic Press, London, 2001).
15. K. G. Wilson, Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture, Phys. Rev. B4, 3174 (1971).
16. K. G. Wilson, Renormalization group and critical phenomena. II. Phase-space cell analysis of critical behavior, Phys. Rev. B4, 3184 (1971).
17. P. Pfeuty and G. Toulouse, Introduction to the Renormalization Group and Critical Phenomena (John Wiley & Sons, Chichester, 1994).
18. M. Plischke and B. Bergersen, Equilibrium Statistical Physics (World Scientific, Singapore, 1994).
19. J. M. Yeomans, Statistical Mechanics of Phase Transitions (Clarendon, Oxford, 1992).
20. K. G. Wilson and J. Kogut, The renormalization group and the ε expansion, Phys. Rep. 12C, 75 (1974).
21. M. E. Fisher, The renormalization group in the theory of critical behavior, Rev. Mod. Phys. 46, 597 (1974).
22. F. J. Wegner, The Critical State, General Aspects in Phase Transitions and Critical Phenomena, Vol. 6, eds. C. Domb and M. S. Green (Academic Press, London, 1976).
23. H. K. Janssen, On the nonequilibrium phase transition in reaction-diffusion systems with an absorbing stationary state, Z. Phys. B42, 151 (1981).
24. P. Grassberger, On phase transitions in Schlögl's second model, Z. Phys. B47, 365 (1982).
25. E. Landau, Handbuch über die Lehre von der Verteilung der Primzahlen (Teubner, Leizig, 1909).
26. M. Abramowitz and I. A. Stegun, Hand Book of Mathematical Functions (Dover, New York, 1973).
27. L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65, 117 (1944).
28. C. N. Yang and T. D. Lee, Statistical theory of equation of state and phase transition: I. Theory of condensation, Phys. Rev. 87, 404 (1952).
29. T. D. Lee and C. N. Yang, Statistical theory of equation of state and phase transition: II. Lattice gas and Ising model, Phys. Rev. 87, 410 (1952).
30. L. D. Landau and E. M. Lifschitz, Lehrbuch der theoretischen Physik, Band V (Akademie, Berlin, 1966).
31. J. W. Essam and M. E. Fisher, Padé approximant studies of lattice gas and Ising ferromagnet below critical point, J. Chem. Phys. 38, 802 (1963).
32. C. Domb and D. L. Hunter, On critical behavior of ferromagnets, P. Phys. Soc. Lond. 86, 1147 (1965).
33. B. Widom, Equation of state in the neighborhood of the critical point, J. Chem. Phys. 43, 3898 (1965).
34. L. P. Kadanoff, The introduction of the idea that exponents could be derived from real-space scaling arguments, Physics (NY) 2, 263 (1966).
35. A. Z. Patashinskii and V. L. Pokrovskii, Behavior of ordered systems near transition point, Sov. Phys. JETP 23, 292 (1966).
36. R. B. Griffiths, Thermodynamic functions for fluids and ferromagnets near the critical point, Phys. Rev. 158, 176 (1967).
37. A. Hankey and H. E. Stanley, Systematic application of generalized homogeneous functions to static scaling, dynamical scaling, and universality, Phys. Rev. B6, 3515 (1972).
38. M. E. Fisher, Quantum corrections to critical-point behavior, Phys. Rev. Lett. 16, 11 (1966).
39. D. Jasnow and M. Wortis, High-temperature critical indices for the classical anisotropic Heisenberg model, Phys. Rev. 176, 739 (1968).
40. P. G. Watson, Formation of invariants from critical amplitudes of ferromagnets, J. Phys. C2, 1883 (1969).
41. R. B. Griffiths, Dependence of critical indices on a parameter, Phys. Rev. Lett. 24, 1479 (1970).
42. D. D. Betts, A. J. Guttmann and G. S. Joyce, Lattice-lattice scaling and the generalized law of corresponding states, J. Phys. C4, 1994 (1971).
43. J. Hubbard and P. Schofield, Wislon theory of a liquid-vapour critical point, Phys. Lett. A40, 245 (1972).
44. N. V. Brilliantov, Effective magnetic Hamiltonian and Ginzburg criterion for fluids, Phys. Rev. E58, 2628 (1998).
45. R. Hocken and M. R. Moldover, Ising critical exponents in real fluids: An experiment, Phys. Rev. Lett. 37, 29 (1976).
46. A. Levelt Sengers, R. Hocken and J. V. Sengers, Critical-point universality and fluids, Physics Today 28, 42 (December) (1977).
47. J. V. Sengers and J. M. H. Levelt Sengers, Critical Phenomena in Classical Fluids in Progress in Liquid Physics, ed. C. A. Croxton (John Wiley & Sons, New York, 1978).
48. J. M. H. Levelt Sengers, From van der Waals' equation to the scaling law, Physica A73, 73 (1974).
49. D. Balzarini and K. Ohrn, Coexistence curve of sulfur hexafluoride, Phys. Rev. Lett. 28, 840 (1972).
50. J. Als-Nielsen, Neutron Scattering and Spatial Correlation Near the Critical Point in Phase Transitions and Critical Phenomena, Vol. 5a, eds. C. Domb and M. S. Green (Academic Press, London, 1976).
51. W, Phys. Rev. Lett. 77, 2887 (1996).
52. B. I. Halperin, T. C. Lubensky and S.-K. Ma, First-order phase transitions in superconductors and smectic-A liquid crystals, Phys. Rev. Lett. 32, 292 (1974).
53. B. I. Halperin and T. C. Lubensky, On the analogy between smetic-A liquid crystals and superconductors, Solid State Comm. 14, 997 (1974).
54. P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University, Cambridge, 1995).
55. 4He in Quantum Liquids, eds. J. Ruvalds and T. Regge (North-Holland, Amsterdam, 1978).
56. J. A. Lipa et al., Heat capacity and thermal relaxation of bulk Helium very near the lambda point, Phys. Rev. Lett. 76, 944 (1996).
57. J. A. Lipa et al., Specific heat of Helium confined to a 57 m planar geometry near the lambda point, Phys. Rev. Lett. 84, 4894 (2000).
58. V. Dohm, Renormalization-group flow equations of model F, Phys. Rev. B44, 2697 (1991).
59. S.-C. Zhang, A unified theory based on SO(5) symmetry of superconductivity and antiferromagnetism, Science 275, 1089 (1997).
60. E. Demler, W. Hanke and S.-C. Zhang, SO(5) theory of antiferromagnetism and superconductivity, Rev. Mod. Phys. 76, 909 (2004).
61. 3He, J. Phys. C9, 743 (1976).
62. G. P. de Gennes, Scaling Concepts on Polymer Physics (Cornell University, Ithaca, 1979).
63. T. H. Berlin and M. Kac, The spherical model of a ferromagnet, Phys. Rev. 86, 821 (1952).
64. H. E. Stanley, Spherical model as the limit of infinite spin dimensionality, Phys. Rev. 176, 718 (1968).
65. G. S. Joyce, Critical Properties of the Spherical Model in Phase Transitions and Critical Phenomena, Vol. 2, eds. C. Domb and M. S. Green (Academic Press, London, 1972).
66. V. Privman and M. E. Fisher, Universal critical amplitudes in finite-size scaling, Phys. Rev. B30, 322 (1984).
67. S. Milošević and H. E. Stanley, Equation of state near the critical point: I. Calculation of the scaling function for S = 1/2 and S = ∞ Heisenberg models using high-temperature series-expansion, Phys. Rev. B6, 986 (1972).
68. V. Privman, P. C. Hohenberg and A. Aharony, Universal Critical-Point Amplitude Relations in Phase Transitions and Critical Phenomena, Vol. 14, eds. by C. Domb and J. L. Lebowitz (Academic Press, London, 1991).
69. A. N. Berker and S. Ostlund, Renormalisation-group calculations of finite systems: Order parameter and specific heat for epitaxial ordering, J. Phys. C12, 4961 (1979).
70. M. Kaufmann and R. B. Griffiths, Exactly soluble Ising models on hierarchical lattices, Phys. Rev. B24, 496 (1981).
71. A. A. Migdal, Phase transitions in gauge and spin-lattice systems, Sov. Phys. JETP 42, 743 (1976).
72. L. P. Kadanoff, Notes on Migdal's recursion formula, Ann. Phys. (NY) 100, 359 (1976).
73. E. Brézin, J. C. Le Guillou and J. Zinn-Justin, Field Theoretical Approach to Critical Phenomena in Phase Transitions and Critical Phenomena, Vol. 6, eds. C. Domb and M. S. Green (Academic Press, London, 1976).
74. F. J. Wegner and E. K. Riedel, Logarithmic corrections to the molecular-field behavior of critical and tricritical systems, Phys. Rev. B7, 248 (1973).
75. F. J. Wegner, Corrections to scaling laws, Phys. Rev. B5, 4529 (1972).
76. M. E. Fisher, Renormalization Group in Critical Phenomena and Quantum Field Theory, eds. D. J. Gunton and M. S. Green (Temple University, Philadelphia, 1974).
77. V. Privman and M. E. Fisher, Finite-size effects at first-order transitions, J. Stat. Phys. 33, 385 (1983).
78. J. G. Brankov, D. M. Danchev and N. S. Tonchev, Theory of critical phenomena in finite-size systems, in Series in Modern Condensed Matter Physics, World Scientific 9 (2000).
79. E. Brézin, An investigation of finite size scaling, J. Phys. (France) 43, 15 (1982).
80. K. G. Wilson and M. E. Fisher, Critical exponents in 3.99 dimensions, Phys. Rev. Lett. 28, 240 (1972).
81. D. J. Walace, The ε-Expansion for Exponents and the Equation of State in Isotropic Systems in Phase Transitions and Critical Phenomena, Vol. 6, eds. C. Domb and M. S. Green (Academic Press, London, 1976).
82. E. Brézin, D. J. Wallace and K. G. Wilson, Feynman-graph expansion for the equation of state near the critical point, Phys. Rev. B7, 232 (1973).
83. c, Phys. Rev. Lett. 31, 1409 (1973).
84. J. F. Nicoll and P. C. Albright, Crossover functions by renormalization-group matching: Three-loop results, Phys. Rev. B31, 4576 (1985).
85. G. Delfino, Universal amplitude ratios in the two-dimensional Ising model, Phys. Lett. B419, 291 (1998).
86. D. Stauffer, Scaling theory of percolation clusters, Phys. Rep. 54, 1 (1979).
87. J. W. Essam, Percolation theory, Rep. Prog. Phys. 43, 833 (1980).
88. D. Stauffer and A. Aharony, Introduction to Percolation Theory (Taylor & Francis, London, 1992).
89. M. Kadar, G. Parisi and Y.-C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Letts. 56, 889 (1986).
90. M. Kadar, Nonequilibrium dynamics of interfaces and lines, Phys. Rep. 301, 85 (1998).
91. A. L. Barabási and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, 1995).
92. T. A. Witten and L. M. Sander, Diffusion limited aggregation: A kinetic critical phenomenon, Phys. Rev. Lett. 47, 1400 (1981).
93. A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, C. R. Acad. Sci. USSR, 30, 299 (1941).
94. U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, 1995).
95. G. Grinstein and M. A. Muñoz, The statistical mechanics of absorbing states, in Lecture Notes in Physics, Springer 493, 223 (1997).
96. E. V. Albano, Critical behavior of a forest fire model with immune trees, J. Phys. A27, L881 (1994).
97. S. Clar, B. Drossel and F. Schwabl, Forest fires and other examples of self-organized criticality, J. Phys.: Condens. Matter 8, 6803 (1996).
98. D. Mollison, Spatial contact models for ecological and epidemic spread, J. Roy. Stat. Soc. B39, 283 (1977).
99. R. M. Ziff, E. Gulari and Y. Barshad, Kinetic phase transitions in an irreversible surface-reaction model, Phys. Rev. Lett. 56, 2553 (1986).
100. Y. Pomeau, Front motion, metastability and subcritical bifurcations in hydrodynamics, Physica D23, 3 (1986).
101. G. I. Menon, S. Sinha and P. Ray, Persistence at the onset of spatiotemporal intermittency in coupled map lattices, Europhys. Lett. 61, 40 (2003).
102. P. Rupp, R. Richter and I. Rehberg, Critical exponents of directed percolation measured in spatiotemporal intermittency, Phys. Rev. E67, 036209 (2003).
103. J. L. Cardy and R. L. Sugar, Directed percolation and Reggeon field theory, J. Phys. A13, L423 (1980).
104. O. Narayan and A. A. Middleton, Avalanches and the renormalization group for pinned charge-density waves, Phys. Rev. B49, 244 (1994).
105. P. Bak, C. Tang and K. Wiesenfeld, Self-organized criticality: An explanation of 1/f noise, Phys. Rev. Lett. 59, 381 (1987).
106. P. Bak, C. Tang and K. Wiesenfeld, Self-organized criticality, Phys. Rev. A38, 364 (1988).
107. P. Bak, How Nature Works (Springer, New York, 1996).
108. D. L. Turcotte, Self-organized criticality, Rep. Prog. Phys. 62, 1377 (1999).
109. S. A. Kaufmann, Emergent properties in random complex automata, Physica D10, 145 (1984).
110. H. J. Herrmann, Damage spreading, Physica A168, 516 (1990).
111. H. Hinrichsen, R. Livi, D. Mukamel and A. Politi, A model for nonequilibrium wetting transitions in two dimensions, Phys. Rev. Lett. 79, 2710 (1997).
112. K. Binder and D. W. Heermann, Monte Carlo Simulation in Statistical Physics (Springer, Berlin, 1997).
113. T. E. Harris, Contact interactions on a lattice, Ann. Prob. 2, 969 (1974).
114. J. Marro and R. Dickman, Nonequilibrium Phase Transitions in Lattice Models (Cambridge University Press, Cambridge, 1999).
115. C. W. Gardiner, K. J. McNeil, D. F. Walls and I. S. Matheson, Correlations in stochastic theories of chemical reactions, J. Stat. Phys. 14, 307 (1976).
116. H. Mori and K. J. McNeil, Critical dimensionality for normal fluctuations of macrovariables in nonequilibrium states, Prog. Theor. Phys. 57, 770 (1977).
117. D. Elderfield and D. D. Vvedensky, Non-equilibrium scaling in the Schlögl model, J. Phys. A18, 2591 (1985).
118. M. Henkel, Local scale invariance and strongly anisotropic equilibrium critical systems, Phys. Rev. Lett. 78, 1940 (1997).
119. →-space instability on the lambda line, Phys. Rev. Lett. 35, 1678 (1975).
120. A. Aharony, Spatially Modulated Structures in Systems with Competing Interactions in Phase Transitions and Critical Phenomena, Vol. 15, eds. C. Domb and J. L. Lebowitz (Academic Press, New York, 1992).
121. A. Hucht, On the symmetry of universal finite-size scaling functions in anisotropic systems, J. Phys. A35, L481 (2002).
122. J. W. Essam, K. de Bell, J. Adler and F. M. Bhatti, Analysis of extended series for bond percolation on the directed square lattice, Phys. Rev. B33, 1982 (1986).
123. J. Adler and J. A. M. S. Duarte, Directed percolation: Field exponents and a test of scaling in two and three dimensions, Phys. Rev. B35, 7046 (1987).
124. M. Adler, J. Berger, J. A. M. S. Duarte and Y. Meir, Directed percolation in 3 + 1 dimensions, Phys. Rev. B37, 7529 (1988).
125. J. W. Essam, A. J. Guttmann and K. de Bell, On two-dimensional directed percolation, J. Phys. A21, 3815 (1988).
126. I. Jensen and R. Dickman, Time-dependent perturbation theory for nonequilibrium lattice models, J. Stat. Phys. 71, 89 (1993).
127. I. Jensen and R. Dickman, Time-dependent perturbation theory for diffisive non-equilibrium lattice models, J. Phys. A26, L151 (1993).
128. I. Jensen and A. J. Guttmann, Series expansions of the percolation probability for directed square and honeycomb lattices, J. Phys. A28, 4813 (1995).
129. I. Jensen, Temporally disordered bond percolation on the directed square lattice, Phys. Rev. Lett. 77, 4988 (1996).
130. I. Jensen and A. J. Guttmann, Series expansions for two-dimensional directed percolation, Nucl. Phys. B (Proc. Suppl.) 47, 835 (1996).
131. I. Jensen, Low-density series expansions for directed percolation on square and triangular lattices, J. Phys. A29, 7013 (1996).
132. I. Jensen, Low-density series expansions for directed percolation, I. A new efficient algorithm with applications to the square lattice, J. Phys. A32, 5233 (1999).
133. I. Jensen, Low-density series expansions for directed percolation, III. Some two-dimensional lattices, J. Phys. A37, 6899 (2004).
134. I. Jensen, Critical behavior of the pair contact process, Phys. Rev. Lett. 70, 1465 (1993).
135. I. Jensen and R. Dickman, Nonequilibrium phase transitions in systems with infinitely many absorbing states, Phys. Rev. E48, 1710 (1993).
136. J. F. F. Mendes, R. Dickman, M. Henkel and M. C. Marques, Generalized scaling for models with multiple absorbing states, J. Phys. A27, 3019 (1994).
137. S. Lübeck, Scaling behavior of the absorbing phase transition in a conserved lattice gas around the upper critical dimension, Phys. Rev. E64, 016123 (2001).
138. S. Lübeck, Scaling behavior of the order parameter and its conjugated field in an absorbing phase transition around the upper critical dimension, Phys. Rev. E65, 046150 (2002).
139. S. Lübeck, Scaling behavior of the conserved threshold transfer process, Phys. Rev. E66, 046114 (2002).
140. S. Lübeck and R. D. Willmann, Universal scaling behaviour of directed percolation and the pair contact process in an external field, J. Phys. A35, 10205 (2002).
141. S. Lübeck and P. C. Heger, Universal scaling behavior at the upper critical dimension of non-equilibrium continuous phase transitions, Phys. Rev. Lett. 90, 230601 (2003).
142. S. Lübeck and R. D. Willmann, Scaling behavior of the directed percolation universality class, J. Stat. Phys. 115, 1231 (2004).
143. S. Lübeck and R. D. Willmann Scaling Behavior of the Directed Percolation Universality Class, in preparation (2005).
144. P. Grassberger and A. de la Torre, Reggeon field theory (Schogl's first model) on a lattice: Monte Carlo calculation of critical behavior, Ann. Phys. (N.Y.) 122, 373 (1979).
145. P. Grassberger, Directed percolation in 2+1 dimensions, J. Phys. A22, 3673 (1989).
146. I. Jensen, H. C. Fogedby and R. Dickman, Critical exponents for an irreversible surface reaction model, Phys. Rev. A41, 3411 (1990).
147. R. Dickman, Nonequilibrium critical behavior of the triplet annihilation model, Phys. Rev. A42, 6985 (1990).
148. I. Jensen, Universality class of a one-dimensional cellular automaton, Phys. Rev. A43, 3187 (1991).
149. I. Jensen, Critical behavior of the three-dimensional contact process, Phys. Rev. A45, R563 (1992).
150. G. Ódor and A. Szolnoki, Directed percolation conjecture for cellular automata, Phys. Rev. E53, 2231 (1996).
151. C. A. Voigt and R. M. Ziff, Epidemic analysis of the second-order transition in the Ziff-Gulari-Barshad surface-reaction model, Phys. Rev. E56, R6241 (1997).
152. d=3 = 0.61 ± 0.02.
153. G. Ódor, Critical behavior in reaction-diffusion systems exhibiting absorbing phase transition, Braz. J. Phys. 33, 431 (2003).
154. I. Jensen, Critical exponents for branching annihilating random walks with an even number of offspring, Phys. Rev. E50, 3623 (1994).
155. J. D. Noh and H. Park, Universality class of absorbing phase transitions with continuously varying exponents, Phys. Rev. E69, 016122 (2004).
156. E. K. Riedel and F. J. Wegner, Effective critical and tricritical exponents, Phys. Rev. B9, 294 (1974).
157. R. Dickman, Numerical study of a field theory for directed percolation, Phys. Rev. E50, 4404 (1994).
158. C. López and M. A. Muñoz, Numerical analysis of a Langevin equation for systems with infinite absorbing states, Phys. Rev. E56, 4864 (1997).
159. J. J. Ramasco, M. A. Muñoz and C. A. da Silva Santos, Numerical study of the Langevin theory for fixed-energy sandpiles, Phys. Rev. E69, 045105 (2004).
160. S. R. Broadbent and J. M. Hammersley, Percolation processes: I. crystals and mazes, Proc. Camb. Phil. Soc. 53, 629 (1957).
161. R. B. Potts, Some generalized order-disorder transformations, Proc. Camb. Phil. Soc. 48, 106 (1952).
162. F. U. Wu, The Potts model, Rev. Mod. Phys. 54, 235 (1982).
163. C. M. Fortuin and P. W. Kasteleyn, On the random cluster model. I. Introduction and relation to other models, Physica 57, 536 (1972).
164. C. M. Fortuin, On the random cluster model. II. The percolation model, Physica 58, 393 (1972).
165. M. P. M. den Nijs, A relation between the temperature exponents of the eight-vertex and q-state Potts model, J. Phys. A12, 1857 (1979).
166. B. Nienhuis, E. K. Riedel and M. Schick, Magnetic exponents of the two-dimensional q-state Potts model, J. Phys. A13, L189 (1980).
167. S. P. Obukhov, The problem of directed percolation, Physcia A101, 145 (1980).
168. H. D. I. Abarbanel, J. B. Bronzan, R. L. Sugar and A. R. White, Reggeon field theory: Formulation and use, Phys. Rep. 21, 119 (1975).
169. T. Ohtsuki and T. Keyes, Nonequilibrium critical phenomena in one-component reaction-diffusion systems, Phys. Rev. A35, 2697 (1987).
170. T. Ohtsuki and T. Keyes, Crossover in nonequilibrium multicritical phenomena of reaction-diffusion systems, Phys. Rev. A36, 4434 (1987).
171. M. Chaichian and A. Demichev, Path Integrals in Physics, Volume 1 (Institute of Physics, Bristol, 2001).
172. H. K. Janssen, Directed percolation with colors and flavors, J. Stat. Phys. 103, 801 (2001).
173. C. J. De Dominicis, Techniques de renormalisation de la théorie des champs et dynamique des phénomènes critiques, J. Phys. C (France) 37, 247 (1976).
174. H. K. Janssen, On a Lagrangean for classical field dynamics and renormalization group calculations of dynamical critical properties, Z. Phys. B23, 377 (1976).
175. P. C. Martin, E. D. Siggia and H. A. Rose, Statistical dynamics of classical systems, Phys. Rev. A8, 423 (1973).
176. H. K. Janssen, Survival and Percolation Probabilities in the Field Theory of Growth Models, cond-mat/0304631 (2004).
177. T. M. Liggett, Interacting Particle Systems (Springer, New York, 1985).
178. J. B. Bronzan and J. W. Dash, Higher-order epsilon terms in Reggeon field theory, Phys. Rev. D10, 4208 (1974).
179. J. B. Bronzan and J. W. Dash, Erratum: Higher-order epsilon terms in Reggeon field theory, Phys. Rev. D12, 1850 (1974).
180. J. B. Bronzan and J. W. Dash, Higher-order ε-terms in the renormalization group approach to Reggeon field theory, Phys. Rev. D12, 1850 (1974).
181. H. K. Janssen, Ü. Kutbay and K. Oerding, Equation of state for directed percolation, J. Phys. A32, 1809 (1999).
182. E. Domany and W. Kinzel, Equivalence of cellular automata to Ising models and directed percolation, Phys. Rev. Lett. 53, 311 (1984).
183. R. Durrett, On the growth of the one dimensional contact processes, Ann. Prob. 8, 890 (1980).
184. R. Durrett and D. Griffeath, On the growth of the one dimensional contact processes, Z. für Wahr. 59, 535 (1982).
185. C. Brezuidenhout and G. Grimmett, The critical contact process dies out, Ann. Prob. 18, 1462 (1990).
186. W. Kinzel, Phase transition of cellular automata, Z. Phys. B58, 229 (1985).
187. T. Tomé, Spreading of the damage in the Domany-Kinzel cellular automaton: A mean-field approach, Physica A212, 99 (1994).
188. H. Rieger, A. Schadschneider and M. Schreckenberg, Re-entrant behaviour in the Domany-Kinzel cellular automaton, J. Phys. A27, L423 (1994).
189. A. P. F. Atman, R. Dickman and J. G. Moreira, Phase diagram of a probabilistic cellular automaton with three-site interactions, Phys. Rev. E67, 016107 (2003).
190. G. F. Zebende and T. J. P. Penna, The Domany-Kinzel cellular automaton phase diagram, J. Stat. Phys. 74, 1273 (1994).
191. J. W. Essam, Directed compact percolation: Cluster size and hyperscaling, J. Phys. A22, 4927 (1989).
192. M. J. Kearney and R. J. Martin, On the finite-size scaling of clusters in compact directed percolation, J. Phys. A: Math. Gen. 36, 6629 (2003).
193. P. Grassberger and Y.-C. Zhang, Self-organized formulation of standard percolation phenomena, Physica A224, 169 (1996).
194. E. Carlon, M. Henkel and U. Schollwöck, Critical properties of the reaction-diffusion model 2A → 3A, 2A → 0, Phys. Rev. E63, 036101 (2001).
195. R. Dickman, Nonuniversality and Critical Point Shift in Systems with Infinitely Many Absorbing Configurations, cond-mat/9909347 (1999).
196. R. Dickman and J. Kamphorst Leal da Silva, Moment ratios for absorbing-state phase transitions, Phys. Rev. E58, 4266 (1998).
197. R. Dickman, W. Rabelo and G. Ódor, Pair contact process with a particle source, Phys. Rev. E65, 016118 (2001).
198. M. A. Muñoz, G. Grinstein, R. Dickman and R. Livi, Critical behavior of systems with many absorbing states, Phys. Rev. Lett. 76, 451 (1996).
199. F. van Wijland, Universality class of nonequilibrium phase transitions with infinitely many absorbing states, Phys. Rev. Lett. 89, 190602 (2002).
200. F. van Wijland, Infinitely-many absorbing-state nonequilibrium phase transitions, Braz. J. Phys. 33, 551 (2003).
201. P. Grassberger, On the critical behavior of the general epidemic process and dynamical percolation, Math. Biosci. 63, 157 (1983).
202. J. L. Cardy, Field theoretic formulation of an epidemic process with immunisation, J. Phys. A16, L706 (1983).
203. J. L. Cardy and P. Grassberger, Epidemic models and percolation, J. Phys. A18, L267 (1985).
204. H. K. Janssen, Renormalized field theory of dynamical percolation, Z. Phys. B58, 311 (1985).
205. J. Kamphorst Leal da Silva and R. Dickman, The pair contact process in two dimensions, Phys. Rev. E60, 5126 (1999).
206. M. J. Howard and U. C. Täuber, Real versus imaginary noise in diffusion-limited reactions, J. Phys. A30, 7721 (1997).
207. G. Ódor, Critical behaviour of the 1d annihilation fission process 2A → 0, 2A → 3A, Phys. Rev. E62, 3027 (2000).
208. H. Hinrichsen, Pair contact process with diffusion: A new type of nonequilibrium critical phenomena?, Phys. Rev. E63, 036102 (2001).
209. H. Hinrichsen, Cyclically coupled spreading and pair annihilation, Physica A291, 275 (2001).
210. G. Ódor, Critical behavior of the one-dimensional diffusive pair contact process, Phys. Rev. E67, 016111 (2003).
211. R. Dickman and M. A. F. de Menezes, Nonuniversality in the pair contact process with diffusion, Phys. Rev. E66, 045101 (2002).
212. G. Ódor, M. C. Marques and M. A. Santos, Phase transition of a two-dimensional binary spreading model, Phys. Rev. E65, 056113 (2002).
213. K. Park and I.-M. Kim, Well-defined set of exponents for a pair contact process with diffusion, Phys. Rev. E66, 027106 (2002).
214. J. Kockelkoren and H. Chaté, Absorbing phase transitions of branching-annihilating random walks, Phys. Rev. Lett. 90, 125701 (2003).
215. G. T. Barkema and E. Carlon, Universality in the pair contact process with diffusion, Phys. Rev. E68, 036113 (2003).
216. H. Hinrichsen, The Diffusive Pair Contact Process and Non-Equilibrium Wetting, cond-mat/0302381 (2003).
217. M. Paessens and G. M. Schütz, Phase transitions and correlations in the bosonic pair contact process with diffusion: Exact results, J. Phys. A37, 4709 (2004).
218. H. K. Janssen, F. van Wijland, O. Deloubrière and U. C. Täuber, Pair contact process with diffusion: Failure of master equation field theory, Phys. Rev. E70, 056114 (2004).
219. M. Henkel and H. Hinrichsen, The non-equilibrium phase transition of the pair-contact process with diffusion, J. Phys. A37, R117 (2004).
220. G. Grinstein, Z. W. Lai and R. J. Browne, Critical phenomena in a nonequilibrium model of heterogeneous catalysis, Phys. Rev. A40, 4820 (1989).
221. D.-J. Liu, N. Pavlenko and J. W. Evans, Crossover between mean-field and Ising critical behavior in a lattice-gas reaction-diffusion model, J. Stat. Phys. 114, 101 (2004).
222. D. Stauffer, M. Ferer and M. Wortis, Universality of second-order phase transitions: The scale factor for the correlation length, Phys. Rev. Lett. 29, 345 (1972).
223. H. K. Janssen and O. Stenull, Logarithmic corrections in directed percolation, Phys. Rev. E69, 016125 (2004).
224. P. Schofield, Parametric representation of the equation of state near a critical point, Phys. Rev. Lett. 22, 606 (1969).
225. B. D. Josephson, Equation of state near the critical point, J. Phys. C2, 1113 (1969).
226. S. P. Obukhov, The problem of directed percolation, Physica A101, 145 (1980).
227. A. I. Larkin and D. E. Khmel'nitskiǐ, Phase transition in uniaxial ferroelectrics, JETP 29, 1123 (1969).
228. A. Aharony, Critical behavior of magnets with dipolar interactions. V. Uniaxial magnets in d dimensions, Phys. Rev. B8, 3363 (1973).
229. 4, Phys. Rev. Lett. 38, 251 (1977).
230. 3, Phys. Rev. Lett. 40, 1286 (1978).
231. N. Aktekin, The finite-size scaling functions of the four-dimensional Ising model, J. Stat. Phys. 104, 1397 (2001).
232. P. Grassberger, Pruned-enriched Rosenbluth method: Simulations of theta polymers of chain length up to 1000000, Phys. Rev. E56, 3682 (1997).
233. P. Grassberger, R. Hegger and L. Schäfer, Self-avoiding walks in four dimensions: Logarithmic corrections, J. Phys. A27, 7265 (1994).
234. A. L. Owczarek and T. Prellberg, Scaling of self-avoiding walks in high dimensions, J. Phys. A34, 5773 (2001).
235. S. Lübeck, Logarithmic corrections of the avalanche distributions of sandpile models at the upper critical dimension, Phys. Rev. E58, 2957 (1998).
236. D. V. Ktitarev, S. Lübeck, P. Grassberger and V. B. Priezzhev, Scaling of waves in the Bak-Tang-Wiesenfeld sandpile model, Phys. Rev. E61, 81 (2000).
237. L. Roters, S. Lübeck and K. D. Usadel, Depinning transition of a driven interface in the random-field Ising model around the upper critical dimension, Phys. Rev. E66, 069901 (2002).
238. P. Grassberger, Critical percolation in high dimensions, Phys. Rev. 67, 036101 (2003).
239. A. A. Fedorenko and S. Stepanow, Depinning transition at the upper critical dimension, Phys. Rev. E67, 057104 (2003).
240. H. K. Janssen and O. Stenull, Logarithmic corrections in dynamical isotropic percolation, Phys. Rev. E68, 036131 (2003).
241. O. Stenull and H. K. Janssen, Logarithmic corrections to scaling in critical percolation and random resistor networks, Phys. Rev. E68, 036129 (2003).
242. D. Grüneberg and A. Hucht, Universal finite-size scaling analysis of Ising models with long-range interactions at the upper critical dimensionality: Isotropic case, Phys. Rev. E69, 036104 (2004).
243. 4- theory at the upper critical dimension, Nucl. Phys. B691, 292 (2004).
244. P. Grassberger, Logarithmic Corrections in (4+1)-Dimensional Directed Percolation, (2004).
245. M. Henkel and U. Schollwöck, Universal finite-size scaling amplitudes in anisotropic scaling, J. Phys. A34, 3333 (2001).
246. J. L. Cardy (ed.), Finite-size scaling, North-Holland, New York, 1988.
247. M. E. Fisher, Theory of critical point singularities, Proceedings of the 1970 Varenna Summer School on Critical Phenomena, ed. M. S. Green (Academic Press, New York, 1971).
248. M. E. Fisher and M. N. Barber, Scaling theory for finite-size effects in the critical region, Phys. Rev. Lett. 28, 1516 (1972).
249. A. D. Bruce, Probability density functions for collective coordinates in Ising-like systems, J. Phys. C14, 3667 (1981).
250. A. D. Bruce, Universality in the two-dimensional continuous spin model, J. Phys. A18, L873 (1985).
251. D. Nicolaides and A. D. Bruce, Universal configurational structure in two-dimensional scalar models, J. Phys. A21, 233 (1988).
252. A. D. Bruce and N. B. Wilding, Scaling fields and universality of the liquid-gas critical point, Phys. Rev. Lett. 68, 193 (1992).
253. C.-K. Hu, C.-Y. Lin and J.-A. Chen, Universal scaling functions in critical phenomena, Phys. Rev. Lett. 75, 193 (1995).
254. K. Kaneda, Y. Okabe and M. Kikuchi, Shape effects of finite-size scaling functions for anistropic three-dimensional Ising models, J. Phys. A32, 7263 (1999).
255. K. Kaneda and Y. Okabe, Finite-size scaling for the Ising model on the Möbius strip and the Klein bottle, Phys. Rev. Lett. 86, 2134 (2001).
256. T. Antal, M. Droz and Z. Rácz, Probability distribution of magnetization in the one-dimensional Ising model: Effects of boundary conditions, J. Phys. A37, 1465 (2004).
257. K. Binder, Finite size scaling analysis of Ising model block distribution functions, Z. Phys. B43, 119 (1981).
258. E. Luijten and H. W. J. Blöte, Finite-size scaling and universality above the upper critical dimensionality, Phys. Rev. Lett. 76, 1557 (1996).
259. K. Binder, M. Nauenberg, V. Privman and A. P. Young, Finite-size tests of hyperscaling, Phys. Rev. B31, 1498 (1985).
260. E. Luijten and H. W. J. Blöte, Classical critical behavior of spin models with long-range interactions, Phys. Rev. B56, 8945 (1997).
261. H. W. J. Blöte and E. Luijten, Universality and five-dimensional Ising model, Europhys. Lett. 38, 565 (1997).
262. E. Luijten, K. Binder and H. W. J. Blöte, Finite-size scaling above the upper critical dimension revisited: The case of the five-dimensional Ising model, Eur. Phys. J. B9, 289 (1999).
263. X. S. Chen and V. Dohm, Failure of universal finite-size scaling above the upper critical dimension, Physica A251, 439 (1998).
264. 4 theory above the upper critical dimension, Eur. Phys. J. B5, 529 (1998).
265. 4 theory of finite-size scaling above the upper critical dimension, Phys. Rev. E63, 016113 (2000).
266. D. Stauffer, World records in the size of simulated Ising models, Braz. J. Phys. 30, 787 (2000).
267. H. K. Janssen, private communication (2004).
268. G. S. Joyce, Spherical model with long-range ferromagnetic interactions, Phys. Rev. 146, 349 (1966).
269. G. Stell, Extension of the Ornstein-Zernike theory of the critical region. II, Phys. Rev. B1, 2265 (1970).
270. M. E. Fisher, S.-K. Ma and B. G. Nickel, Critical exponents for long-range interactions, Phys. Rev. Lett. 29, 917 (1972).
271. M. Aizenman and R. Fernández, Critical exponents for long-range interactions, Lett. Math. Phys. 16, 39 (1988).
272. H. K. Janssen, B. Schaub and B. Schmittmann, Finite size scaling for directed percolation and related stochastic evolution processes, Z. Phys. B71, 377 (1988).
273. H. K. Janssen, B. Schaub and B. Schmittmann, New universal short-time scaling behaviour of critical relaxation processes, Z. Phys. B73, 539 (1988).
274. R. Dickman and A. Yu. Tretyakov, Hyperscaling in the Domany-Kinzel cellular automaton, Phys. Rev. E52, 3218 (1995).
275. W. Kinzel, Directed percolation, Ann. Israel Phys. Soc. 5, 425 (1983).
276. F. van Wijland, K. Oerding and H. J. Hilhorst, Wilson renormalization of a reaction-diffusion process, Physica A251, 179 (1998).
277. A. Daerr and S. Douady, Two types of avalanche behavior in granular media, Nature 399, 241 (1999).
278. H. Hinrichsen, Flowing sand: A physical realization of directed percolation, Phys. Rev. Lett. 83, 4999 (1999).
279. H. Hinrichsen, On possible experimental realizations of directed percolation, Braz. J. Phys. 30, 69 (2000).
280. P. Grassberger, Directed Percolation: Results and Open Problems in Nonlinearities in Complex Systems, eds. S. Pury et al. (Narosa Publishing, New Dehli, 1997).
281. C. Sire, Approximation for directed percolation in d = 1 + 1, Phys. Rev. E66, 046133 (2002).
282. D. Zhong and D. ben Avraham, Universality class of two-offspring branching annihilating random walks, Phys. Lett. A209, 333 (1995).
283. J. L. Cardy and U. C. Täuber, Theory of branching and annihilating random walks, Phys. Rev. Lett. 13, 4780 (1996).
284. M. Rossi, R. Pastor-Satorras and A. Vespignani, The universality class of absorbing phase transition with a conserved field, Phys. Rev. Lett. 85, 1803 (2000).
285. R. Dickman, A. Vespignani and S. Zapperi, Self-organized criticality as an absorbing-state phase transition, Phys. Rev. E57, 5095 (1998).
286. R. Pastor-Satorras and A. Vespignani, Field theory of absorbing phase transitions with a nondiffusive conserved field, Phys. Rev. E62, 5875 (2000).
287. J. F. Stilck, R. Dickman and R. R. Vidigal, Series expansion for a stochastic sandpile, J. Phys. A37, 1145 (2004).
288. R. Dickman and R. R. Vidigal, Path-integral representation for a stochastic sandpile, J. Phys. A35, 7269 (2002).
289. S. S. Manna, Two-state model of self-organized criticality, J. Phys. A24, L363 (1991).
290. A. Ben-Hur and O. Biham, Universality in sandpile models, Phys. Rev. E53, R1317 (1996).
291. D. Dhar, Some results and a conjecture for Manna's stochastic sandpile model, Physica A270, 69 (1999).
292. A. Chessa, H. E. Stanley, A. Vespignani and S. Zapperi, Universality in sandpiles, Phys. Rev. E59, 12 (1999).
293. A. Chessa, A. Vespignani and S. Zapperi, Critical exponents in stochastic sandpile models, Comp. Phys. Com. 121, 299 (1999).
294. S. Lübeck, Moment analysis of the probability distributions of different sandpile models, Phys. Rev. E61, 204 (2000).
295. S. Lübeck, Crossover phenomenon in self-organized critical sandpile models, Phys. Rev. E62, 6149 (2000).
296. R. Dickman et al., Critical behavior of a one-dimensional fixed-energy stochastic sandpile, Phys. Rev. E64, 056104 (2001).
297. O. Biham, E. Milshtein and O. Maclai, Evidence for universality within the classes of deterministic and stochastic sandpile models, Phys. Rev. E63, 061309 (2001).
298. Y. Shilo and O. Biham, Sandpile models and random walkers on finite lattices, Phys. Rev. E67, 066102 (2003).
299. R. Kree, B. Schaub and B. Schmitmann, Effects of pollution on critical population dynamics, Phys. Rev. A39, 2214 (1989).
300. R. Pastor-Satorras and A. Vespignani, Reaction-diffusion system with self-organized critical behavior, Eur. Phys. J. B19, 583 (2001).
301. A. Vespignani, R. Dickman, M. A. Muñoz and S. Zapperi, Absorbing state phase transition in fixed-energy sandpiles, Phys. Rev. E62, 4564 (2000).
302. S. Lübeck, Universal behavior of crossover scaling functions for continuous phase transitions, Phys. Rev. Lett. 90, 210601 (2003).
303. P. C. Heger, Computersimulationen zur Bestimmung des Skalenverhaltens absorbierender Phasenübergänge, Diplomarbeit, Universität Duisburg-Essen, 2003.
304. S. Lübeck and A. Hucht, Mean-field scaling function of the universality class of absorbing phase transitions with a conserved field, J. Phys. A35, 4853 (2002).
305. R. Dickman, T. Tomé and M. J. de Oliveira, Sandpiles with height restrictions, Phys. Rev. E66, 016111 (2002).
306. H. J. Jensen, Lattice gas as a model of 1/f noise, Phys. Rev. Lett. 64, 3103 (1990).
307. S. Lübeck and K. D. Usadel, The Bak-Tang-Wiesenfeld sandpile model around the upper critical dimension, Phys. Rev. E56, 5138 (1997).
308. S. Lübeck, Ph.D. thesis, Gerhard-Mercator-Universität Duisburg, 1998.
309. N. Madras and G. Slade, The Self-Avoiding Walk (Birkhäuser, Boston, 1996).
310. G. F. Lawler, A self-avoiding random walk, Duke Math. J. 47, 655 (1980).
311. G. F. Lawler, Gaussian behavior of loop-erased self-avoiding random walk in four dimensions, Duke Math. J. 53, 249 (1986).
312. A. Coniglio, Fractal structure of Ising and Potts clusters: Exact results, Phys. Rev. Lett. 62, 3054 (1989).
313. G. F. Lawler, Intersections of Random Walks (Birkhäuser, Boston, 1991).
314. M. A. Muñoz, R. Dickman, A. Vespignani and S. Zapperi, Avalanche and spreading exponents in systems with absorbing states, Phys. Rev. E59, 6175 (1999).
315. P. Bak and K. Sneppen, Punctuated equilibrium and criticality in a simple model of evolution, Phys. Rev. Lett. 71, 4083 (1993).
316. D. Dhar, Self-organized critical state of sandpile automaton models, Phys. Rev. Lett. 64, 1613 (1990).
317. S. N. Majumdar and D. Dhar, Equivalence between the Abelian sandpile model and the q → 0 limit of the Potts model, Physica A185, 129 (1992).
318. S. N. Majumdar and D. Dhar, Height correlations in the Abelian sandpile model, J. Phys. A24, L357 (1991).
319. V. B. Priezzhev, Structure of the two-dimensional sandpile, J. Stat. Phys 74, 955 (1994).
320. E. V. Ivashkevich, Boundary height correlations in two-dimensional Abelian sandpile, J. Phys. A27, 3643 (1994).
321. E. V. Ivashkevich, D. V. Ktitarev and V. B. Priezzhev, Waves of topplings in an Abelian sandpile, Physica A209, 347 (1994).
322. V. B. Priezzhev, D. V. Ktitarev and E. V. Ivashkevich, Formation of avalanches and critical exponents in an Abelian sandpile model, Phys. Rev. Lett. 76, 2093 (1994).
323. E. V. Ivashkevich, D. V. Ktitarev and V. B. Priezzhev, Critical exponents for boundary avalanches in two-dimensional Abelian sandpile, J. Phys. A27, L585 (1994).
324. V. B. Priezzhev, The upper critical dimension of the Abelian sandpile model, J. Stat. Phys. 98, 667 (2000).
325. S. S. Manna, Large-Scale simulation of avalanche cluster distribution in sandpile model, J. Stat. Phys. 59, 509 (1990).
326. P. Grassberger and S. S. Manna, Some more sandpiles, J. Phys. (France) 51, 1077 (1990).
327. S. Lübeck and K. D. Usadel, Numerical determination of the avalanche exponents of the Bak-Tang-Wiesenfeld model, Phys. Rev. E55, 4095 (1997).
328. M. De Menech, A. L. Stella and C. Tebaldi, Rare events and breakdown of simple scaling in the Abelian sandpile model, Phys. Rev. E58, 2677 (1998).
329. C. Tebaldi, M. De Menech and A. L. Stella, Multifractal scaling in the Bak-Tang-Wiesenfeld Sandpile and edge events, Phys. Rev. Lett. 83, 3952 (1999).
330. G. Grinstein, Generic scale invariance and self-organized criticality, in Scale Invariance, Interfaces, and Non-Equilibrium Dynamics, eds. A. McKane et al., NATO ASI Series B: Physics Vol. 344 (Plenum Press, London, 1995).
331. M. Vergeles, A. Maritan and J. Banavar, Mean-field theory of sandpiles, Phys. Rev. E55, 1998 (1997).
332. A. Vespignani and S. Zapperi, How self-organized criticality works: A unified meanfield picture, Phys. Rev. E57, 6345 (1998).
333. C. Tang and P. Bak, Mean field theory of self-organized critical phenomena, J. Stat. Phys. 51, 797 (1988).
334. A. Vespignani and S. Zapperi, Order parameter and scaling fields in self-organized criticality, Phys. Rev. Lett. 78, 4793 (1997).
335. A. Vespignani, R. Dickman, M. A. Muñoz and S. Zapperi, Driving, conservation, and absorbing states in sandpiles, Phys. Rev. Lett. 81, 5676 (1998).
336. F. Bagnoli, F. Cecconi, A. Flammini and A. Vespignani, Short period attractors and non-ergodic behavior in the deterministic fixed energy sandpile model, Europhys. Lett. 63, 512 (2003).
337. P. Alstrøm, Mean-field exponents for self-organized critical phenomena, Phys. Rev. A38, 4905 (1988).
338. D. Dhar and S. N. Majumdar, Abelian sandpile model on the Bethe lattice, J. Phys. A23, 4333 (1990).
339. S. A. Janowsky and C. A. Laberge, Exact solutions for a mean-field Abelian sandpile, J. Phys. A26, L973 (1993).
340. R. Garćia-Pelayo, I. Salazar and W. C. Schieve, A branching process model for sandpile avalanches, J. Stat. Phys. 72, 167 (1993).
341. S. Zapperi, K. B. Lauritsen and H. E. Stanley, Self-organized branching process: mean-field theory for avalanches, Phys. Rev. Lett. 75, 4071 (1995).
342. E. V. Ivashkevich, Critical behavior of the sandpile model as a self-organized branching process, Phys. Rev. Lett. 76, 3368 (1996).
343. M. Katori and H. Kobayashi, Mean-field theory of avalanches in self-organized critical states, Physica A229, 461 (1996).
344. G. Caldarelli, Mean field theory for ordinary and hot sandpiles, Physica A252, 295 (1998).
345. B. Gutenberg and C. F. Richter, Seismicity of the Earth and Associated Phenomena (Princeton University Press, Princeton, New Jersey, 1954).
346. R. Dickman, M. A. Muñoz, A. Vespignani and S. Zapperi, Paths to self-organized criticality, Braz. J. Phys. 30, 27 (2000).
347. A. Aharony, Dependence of Universal Critical Behavior on Symmetry and Range of Interaction in Phase Transitions and Critical Phenomena, Vol. 6, eds. C. Domb and M. S. Green (Academic Press, London, 1976).
348. E. K. Riedel and F. J. Wegner, Scaling approach to anisotropic magnetic systems statics, Z. Phys. 225, 195 (1969).
349. M.-C. Chang and A. Houghton, Universal ratios among correction-to-scaling amplitudes on the coexistence curve, Phys. Rev. Lett. 44, 785 (1980).
350. E. Luijten, H. W. J. Blöte and K. Binder, Nonmomotonic crossover of the effective susceptibility exponent, Phys. Rev. Lett. 79, 561 (1997).
351. E. Luijten and K. Binder, Nature of crossover from classical to Ising-like critical behavior, Phys. Rev. E58, 4060 (1998).
352. M. I. Marqués and J. A. Gonzalo, Scaling relationship between effective critical exponents throughout the crossover region in thin films, Eur. Phys. J. B14, 317 (2000).
353. S. Lübeck, Violation of the Widom scaling law for effective crossover exponents, Phys. Rev. E69, 066101 (2004).
354. D. S. Greywall and G. Ahlers, Second-sound velocity, scaling, and universality in He II under pressure near the super-fluid transition, Phys. Rev. Lett. 28, 1251 (1972).
355. M. A. Anisimov, S. B. Kiselev, J. V. Sengers and S. Tang, Crossover approach to global critical phenomena in fluids, Physica A188, 487 (1992).
356. M. Y. Belyakov and S. B. Kiselev, Crossover behavior of the susceptibility of the susceptibility and the specific heat near a second-order phase transition, Physica A190, 75 (1992).
357. E. Luijten, H. W. J. Blöte and K. Binder, Medium-range interactions and crossover to classical critical behavior, Phys. Rev. E54, 4626 (1996).
358. K. K. Mon and K. Binder, Finite-size scaling and the crossover to mean-field critical behavior in the two-dimensional Ising model with medium-ranged interactions, Phys. Rev. E48, 2498 (1993).
359. A. Pelissetto, P. Rossi and E. Vicari, Crossover scaling from classical to nonclassical critical behavior, Phys. Rev. E58, 7146 (1998).
360. A. Pelissetto, P. Rossi and E. Vicari, Mean-field expansion for spin models with medium-range interactions, Nucl. Phys. B554, 552 (1999).
361. S. Caracciolo et al., Crossover phenomena in spin models with medium-range interactions and self-avoiding walks with medium-range jumps, Phys. Rev. E64, 046130 (2001).
362. Y. C. Kim, M. A. Anisimov, J. V. Sengers and E. Luijten, Crossover critical behavior in the three-dimensional Ising model, J. Stat. Phys. 110, 591 (2003).
363. V. L. Ginzburg, Some remarks on the phase transitions of the second kind and the microscopic theory of ferroelectric materials, Sov. Phys. Solid State 2, 1824 (1960).
364. J. Als-Nielsen and R. J. Birgeneau, Mean-field theory, the Ginzburg criterion, and marginal dimensionality of phase transitions, Am. J. Phys. 45, 554 (1977).
365. S. Lübeck and A. Hucht, Absorbing phase transition in a conserved lattice gas with random neighbors, J. Phys. A34, L577 (2001).
366. B. Derrida, A. J. Bray and C. Godrèche, Non-trivial exponents in the zero temperature dynamics of the 1D Ising and Potts model, J. Phys. A27, L357 (1994).
367. A. J. Bray, B. Derrida and C. Godrèche, Non-trivial algebraic decay in a soluble model of coarsening, Europhys. Lett. 27, 175 (1994).
368. D. Stauffer, Ising spinodal decomposition at T = 0 in one to five dimensions, J. Phys. A27, 5029 (1994).
369. S. N. Majumdar, A. J. Bray, S. J. Corwell and C. Sire, Global persistence exponent for nonequilibrium critical dynamics, Phys. Rev. Lett. 77, 3704 (1996).
370. S. Cueille and C. Sire, Spin block persistence at finite temperature, J. Phys. A30, L791 (1997).
371. S. N. Majumdar, Persistence in nonequilibrium systems, Curr. Sci. India 77, 370 (1999).
372. B. Derrida, V. Hakim and V. Pasquier, Exact first-passage exponents of 1D domain growth: Relation to a reaction-diffusion model, Phys. Rev. Lett. 75, 751 (1995).
373. B. P. Lee and A. D. Rutenberg, Persistence, poisoning, and autocorrelations in dilute coarsening, Phys. Rev. Lett. 79, 4842 (1997).
374. K. Oerding and F. van Wijland, Global persistence in directed percolation, J. Phys. A31, 7011 (1998).
375. H. Hinrichsen and M. Antoni, Identification of domain walls in coarsening systems at finite temperature, Phys. Rev. E57, 2650 (1998).
376. E. V. Albano and M. A. Muñoz, Numerical study of persistence in models with absorbing states, Phys. Rev. E63, 031104 (2001).
377. H. Hinrichsen and H. M. Koduvely, Numerical study of local and global persistence in directed percolation, Eur. Phys. J. B5, 257 (1998).
378. S. Lübeck and A. Misra, Persistence distributions in a conserved lattice gas with absorbing states, Eur. Phys. J. B26, 75 (2002).
379. H. K. Janssen, Renormalization field theory of the Gribov process with quenched disorder, Phys. Rev. E55, 6253 (1997).
380. A. G. Moreira, Critical dynamics of the contact process with quenched disorder, Phys. Rev. E54, 3090 (1996).
381. R. Cafiero, A. Gabrielli and M. A. Muñoz, Disordered one-dimensional contact process, Phys. Rev. E57, 5060 (1998).
382. J. Hooyberghs, F. Iglói and C. Vanderzande, Absorbing state phase transitions with quenched disorder, Phys. Rev. E69, 066140 (2004).
383. T. Vojta, Broadening of a nonequilibrium phase transition by extended structural defects, Phys. Rev. E70, 026108 (2004).
384. S. Brazovskii and T. Nattermann, Pinning and sliding of driven elastic systems: From domain walls to charge density waves, Adv. Phys. 53, 177 (2004).
385. U. Nowak and K. D. Usadel, Influence of the temperature on the depinning transition of driven interfaces, Europhys. Lett. 44, 634 (1998).
386. L. Roters et al., Depinning transition and thermal fluctuations in the random-field Ising model, Phys. Rev. E60, 5202 (1999).
387. L. Roters, S. Lübeck and K. D. Usadel, Creep motion in a random-field Ising model, Phys. Rev. E63, 026113 (2001).
388. V. Repain et al., Creep motion of a magnetic wall: Avalanche size divergence, Europhys. Lett. 68, 460 (2004).
389. T. E. Harris, The Theory of Branching Processes (Springer, Berlin, 1963).
390. G. R. Grimmet and D. R. Stirzaker, Probability and Random Processes (Clarendon Press, Oxford, 1992).
391. R. Otter, The multiplicative process, Ann. Math. Stat. 20, 206 (1949).
392. R. Dickman, Reweighting in non-equilibrium simulations, Phys. Rev. E60, 2441 (1999).
393. P. Grassberger, private communication (2004).
394. S. Lübeck and H. K. Janssen, Finite-size scaling of directed percolation above the upper critical dimension, submitted for publication, 2005.