An Exactly Solvable Anisotropic Directed Percolation Model in Three Dimensions

R. Rajesh; Deepak Dhar

doi:10.1103/PhysRevLett.81.1646

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An Exactly Solvable Anisotropic Directed Percolation Model in Three Dimensions

R. Rajesh and Deepak Dhar

Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

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Phys. Rev. Lett. 81, 1646 – Published 24 August, 1998

DOI: https://doi.org/10.1103/PhysRevLett.81.1646

Abstract

We solve exactly a special case of the anisotropic directed-bond percolation problem in three dimensions, in which the occupation probability is 1 along two spatial directions, by mapping it to a five-vertex model. We determine the asymptotic shape of the infinite cluster and hence the direction dependent critical probability. The exponents characterizing the fluctuations of the boundary of the wetted cluster in $d$ dimensions are related to those of the $(d - 2)$ -dimensional KPZ equation.

References (17)

P. Grassberger, J. Stat. Phys. 79, 13 (1995).
R. M. Ziff, E. Gulari, and Y. Barshad, Phys. Rev. Lett. 56, 2553 (1986).
S. Maslov and Y. C. Zhang, Physica (Amsterdam) 223A, 1 (1996); B. Tadic and D. Dhar, Phys. Rev. Lett. 79, 1519 (1997).
L. H. Tang and H. Leschhorn, Phys. Rev. A 45, R8309 (1992); S. V. Buldyrev, A. L. Barabási, F. Caserta, S. Havlin, H. E. Stanley, and T. Vicsek, 45, R8313 (1992).
I. Jensen, J. Phys. A 29, 7013 (1996); J. W. Essam, A. J. Guttmann, and K. De'Bell, 21, 3815 (1988); R. J. Baxter and A. J. Guttmann, 21, 3193 (1988); P. Grassberger and Y. C. Zhang, Physica (Amsterdam) 224A, 169 (1996).
E. Domany and W. Kinzel, Phys. Rev. Lett. 47, 5 (1981).
F. Y. Wu and H. E. Stanley, Phys. Rev. Lett. 48, 775 (1982).
A recent discussion of the exponent $θ$ can be found in S. Lai and M. E. Fisher, J. Chem. Phys. 103, 8144 (1995). For undirected percolation, within the droplet model, the value of $θ^{'}$ is known in all dimensions [see T. C. Lubensky and A. J. McKane, J. Phys. A 14, L157 (1981)].
M. Kardar, G. Parisi, and Y. C. Zhang, Phys. Rev. Lett. 56, 889 (1986); T. Halpin-Healy and Y. C. Zhang, Phys. Rep. 254, 216 (1995).
P. Rujan, J. Stat. Phys. 49, 139 (1987); A. Georges and P. Le Doussal, 54, 1011 (1989).
In the case of the off-critical directed animal problem, the dimensional reduction by 1 is discussed in J. L. Cardy, J. Phys. A 15, L593 (1982).
For other soluble models in three dimensions, see V. V. Bazhanov and R. J. Baxter, J. Stat. Phys. 69, 453 (1992); H. Y. Huang, V. Popkov, and F. Y. Wu, Phys. Rev. Lett. 78, 409 (1997); V. B. Priezzhev, J. Stat. Phys. 26, 817 (1981); J. F. Nagle, C. S. O. Yokoi, and S. M. Bhattacharjee, in Phase Transitions and Critical Phenomena, C. Domb and J. L. Lebowitz (Academic Press, London, 1990), Vol. 13; M. Prähofer and H. Spohn, J. Stat. Phys. 88, 999 (1997).
This is a special case of the six-vertex model for which the exact solution is known. For a review of the six-vertex model, see E. H. Lieb and F. Y. Wu, in Phase Transitions and Critical Phenomena, C. Domb and M. S. Green (Academic Press, London, 1972), Vol. 1, pp. 331–490. References to more recent work may be found in H. Y. Huang, F. Y. Wu, H. Kunz, and D. Kim, Physica (Amsterdam) 228A, 1 (1996).
E. H. Lieb and F. Y. Wu, in Phase Transitions and Critical Phenomena (Ref. [[13]]), p. 363.
E. H. Lieb and F. Y. Wu, in Phase Transitions and Critical Phenomena (Ref. [[13]]), pp. 374–375.
D. Dhar, Phase Trans. 9, 51 (1987); L. H. Gwa and H. Spohn, Phys. Rev. A 46, 844 (1992).
L. H. Tang, M. Kardar, and D. Dhar, Phys. Rev. Lett. 74, 920 (1995).

[c1] P. Grassberger, J. Stat. Phys. 79, 13 (1995).

[c2] R. M. Ziff, E. Gulari, and Y. Barshad, Phys. Rev. Lett. 56, 2553 (1986).

[c3] S. Maslov and Y. C. Zhang, Physica (Amsterdam) 223A, 1 (1996); B. Tadic and D. Dhar, Phys. Rev. Lett. 79, 1519 (1997).

[c4] L. H. Tang and H. Leschhorn, Phys. Rev. A 45, R8309 (1992); S. V. Buldyrev, A. L. Barabási, F. Caserta, S. Havlin, H. E. Stanley, and T. Vicsek, 45, R8313 (1992).

[c5] I. Jensen, J. Phys. A 29, 7013 (1996); J. W. Essam, A. J. Guttmann, and K. De'Bell, 21, 3815 (1988); R. J. Baxter and A. J. Guttmann, 21, 3193 (1988); P. Grassberger and Y. C. Zhang, Physica (Amsterdam) 224A, 169 (1996).

[c6] E. Domany and W. Kinzel, Phys. Rev. Lett. 47, 5 (1981).

[c7] F. Y. Wu and H. E. Stanley, Phys. Rev. Lett. 48, 775 (1982).

[c8] A recent discussion of the exponent $θ$ can be found in S. Lai and M. E. Fisher, J. Chem. Phys. 103, 8144 (1995). For undirected percolation, within the droplet model, the value of $θ^{'}$ is known in all dimensions [see T. C. Lubensky and A. J. McKane, J. Phys. A 14, L157 (1981)].

[c9] M. Kardar, G. Parisi, and Y. C. Zhang, Phys. Rev. Lett. 56, 889 (1986); T. Halpin-Healy and Y. C. Zhang, Phys. Rep. 254, 216 (1995).

[c10] P. Rujan, J. Stat. Phys. 49, 139 (1987); A. Georges and P. Le Doussal, 54, 1011 (1989).

[c11] In the case of the off-critical directed animal problem, the dimensional reduction by 1 is discussed in J. L. Cardy, J. Phys. A 15, L593 (1982).

[c12] For other soluble models in three dimensions, see V. V. Bazhanov and R. J. Baxter, J. Stat. Phys. 69, 453 (1992); H. Y. Huang, V. Popkov, and F. Y. Wu, Phys. Rev. Lett. 78, 409 (1997); V. B. Priezzhev, J. Stat. Phys. 26, 817 (1981); J. F. Nagle, C. S. O. Yokoi, and S. M. Bhattacharjee, in Phase Transitions and Critical Phenomena, C. Domb and J. L. Lebowitz (Academic Press, London, 1990), Vol. 13; M. Prähofer and H. Spohn, J. Stat. Phys. 88, 999 (1997).

[c13] This is a special case of the six-vertex model for which the exact solution is known. For a review of the six-vertex model, see E. H. Lieb and F. Y. Wu, in Phase Transitions and Critical Phenomena, C. Domb and M. S. Green (Academic Press, London, 1972), Vol. 1, pp. 331–490. References to more recent work may be found in H. Y. Huang, F. Y. Wu, H. Kunz, and D. Kim, Physica (Amsterdam) 228A, 1 (1996).

[c14] E. H. Lieb and F. Y. Wu, in Phase Transitions and Critical Phenomena (Ref. [[13]]), p. 363.

[c15] E. H. Lieb and F. Y. Wu, in Phase Transitions and Critical Phenomena (Ref. [[13]]), pp. 374–375.

[c16] D. Dhar, Phase Trans. 9, 51 (1987); L. H. Gwa and H. Spohn, Phys. Rev. A 46, 844 (1992).

[c17] L. H. Tang, M. Kardar, and D. Dhar, Phys. Rev. Lett. 74, 920 (1995).

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An Exactly Solvable Anisotropic Directed Percolation Model in Three Dimensions

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