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10
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85036323690
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e-printcond-mat/9909009
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for a review, also see D. Dhar, e-print cond-mat/9909009.
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Dhar, D.1
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17
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85036257386
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e-print cond-mat/9904054
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V.B. Priezzhev, e-print cond-mat/9904054.
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Priezzhev, V.B.1
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19
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0003672584
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A. McKane, Plenum, New York
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G. Grinstein, in Scale Invariance, Interfaces and Nonequilibrium Dynamics, Vol. 344 of NATO Advanced Study Institute, Series B: Physics, edited by A. McKane (Plenum, New York, 1995).
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Scale Invariance, Interfaces and Nonequilibrium Dynamics, Vol. 344 of NATO Advanced Study Institute, Series B: Physics
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Grinstein, G.1
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24
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3843080724
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A. Vespignani, R. Dickman, M.A. Muñoz, and Stefano Zapperi, Phys. Rev. Lett. 81, 5676 (1998).
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Phys. Rev. Lett.
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Vespignani, A.1
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29
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0001470344
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Deviations from criticality with respect to the driving field can be obtained in the case of fast driving. See Ref. 12 and A. Barrat, A. Vespignani, and S. Zapperi, Phys. Rev. Lett. 83, 1962 (1999).
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Phys. Rev. Lett.
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Barrat, A.1
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32
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0000943599
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The question of open vs closed models for SOC was also discussed in A. Montakhab and J.M. Carlson, Phys. Rev. E 58, 5608 (1998).
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Phys. Rev. E
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Montakhab, A.1
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4244215017
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An early study of sandpiles varying the total energy can be found in C. Tang and P. Bak, Phys. Rev. Lett. 60, 2347 (1988).
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Phys. Rev. Lett.
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Tang, C.1
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35
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0001697794
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P. Garrido, J. Marro, Springer-Verlag, Berlin
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G. Grinstein and M.A. Muñoz, in Fourth Granada Lectures in Computational Physics, edited by P. Garrido and J. Marro, Lecture Notes in Physics Vol. 493 (Springer-Verlag, Berlin, 1997), p. 223
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Fourth Granada Lectures in Computational Physics
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Grinstein, G.1
Muñoz, M.A.2
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37
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0030705043
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and references therein
-
See H. Leschhorn, T. Nattermann, S. Stepanow, and L.-H. Tang, Ann. Phys. (N.Y.) 6, 1 (1997), and references therein.
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(1997)
Ann. Phys. (N.Y.)
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Leschhorn, H.1
Nattermann, T.2
Stepanow, S.3
Tang, L.-H.4
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39
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85036353910
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-
Like any other statistical model, a fixed-energy sandpile exhibits critical singularities only in the infinite-size limit. In this limit the activity density is strictly zero for (Formula presented) and positive for (Formula presented) ensuring the stated inequality for (Formula presented) in the slowly driven system
-
Like any other statistical model, a fixed-energy sandpile exhibits critical singularities only in the infinite-size limit. In this limit the activity density is strictly zero for (Formula presented) and positive for (Formula presented) ensuring the stated inequality for (Formula presented) in the slowly driven system.
-
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40
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0032629418
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D. Dhar, Physica A 270, 69 (1999).
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Physica A
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Dhar, D.1
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45
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0002429349
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The connection with RFT theory was also discussed for the Bak-Sneppen SOC model. See S. Maslov, M. Paczuski, and P. Bak, Europhys. Lett. 27, 97 (1994)
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(1994)
Europhys. Lett.
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Maslov, S.1
Paczuski, M.2
Bak, P.3
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53
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0031497016
-
-
This is the inclusive version of the Manna model. It is also possible to define an exclusive version in which the two toppling particles are forbidden to go to the same neighboring site: H. Kobayashi and M. Katori, J. Phys. Soc. Jpn. 66, 2367 (1997).
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(1997)
J. Phys. Soc. Jpn.
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Kobayashi, H.1
Katori, M.2
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55
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0002674992
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A. Chessa, H.E. Stanley, A. Vespignani, and S. Zapperi, Phys. Rev. E 59, R12 (1999).
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Chessa, A.1
Stanley, H.E.2
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Zapperi, S.4
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58
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85036156354
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(private communication)
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R. Pastor-Satorras (private communication).
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Pastor-Satorras, R.1
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59
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85036423742
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(private communication)
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P. Grassberger (private communication).
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Grassberger, P.1
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60
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85036388777
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-
the BTW model, one actually encounters immortal configurations—in which activity never ceases—having (Formula presented) The probability of generating such initial configurations decays rapidly with system size, and in practice we have not seen them for (Formula presented)
-
In the BTW model, one actually encounters immortal configurations—in which activity never ceases—having (Formula presented) The probability of generating such initial configurations decays rapidly with system size, and in practice we have not seen them for (Formula presented).
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63
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0002601152
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-
Interestingly, a set of coupled Langevin equations with thermal noises has been designed to model the evolving surface of more realistic sandpile models. Also in this case the coupling between moving particles and the density of immobile material represents a crucial point of the theory. See A. Mehta, J.M. Luck, and R.J. Needs, Phys. Rev. E 53, 92 (1996)
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Phys. Rev. E
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Mehta, A.1
Luck, J.M.2
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64
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0000484586
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Phys. Rev. EP. Biswas, A. Majumdar, A. Mehta, and J.K. Bhattacharjee, 58, 1266 (1998).
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Phys. Rev. E
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Biswas, P.1
Majumdar, A.2
Mehta, A.3
Bhattacharjee, J.K.4
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65
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85036350001
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-
While our ansatz simplifies an analysis of stationary states, a theory of transient or spreading dynamics may require retaining (Formula presented) as an independent field, if its initial value differs from the stationary one, (Formula presented). (The situation is analogous to that of a “non-natural” initial density in the PCP 83
-
While our ansatz simplifies an analysis of stationary states, a theory of transient or spreading dynamics may require retaining (Formula presented) as an independent field, if its initial value differs from the stationary one, (Formula presented). (The situation is analogous to that of a “non-natural” initial density in the PCP 83.)
-
-
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-
67
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0001387229
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This is the case of branching, annihilating random walks with even numbers of offspring, also known as the “parity conserving” or “directed Ising” universality class. See P. Grassberger, F. Krause, and T. von der Twer, J. Phys. A 17, L105 (1984)
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(1984)
J. Phys. A
, vol.17
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Grassberger, P.1
Krause, F.2
von der Twer, T.3
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74
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0001366270
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Phys. Rev. EW. Hwang, S. Kwon, H. Park, and H. Park, 57, 6438 (1998).
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Phys. Rev. E
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, pp. 6438
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Hwang, W.1
Kwon, S.2
Park, H.3
Park, H.4
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81
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85036322871
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-
the former case, e.g., for a site-diluted contact process 55, activity tends to be restricted to favorable regions (lower than average dilution). In the present case, it is principally at the boundaries between active and inactive regions that, as implied by the Laplacian in Eq. (1), energy is transferred, and the effect is to move energy into the inactive region, thereby enhancing the further spread of activity. Indeed, the simulations reported below reveal none of the hallmarks of quenched disorder in the contact process, such as logarithmic time dependence in critical spreading, or generic power-law relaxation of temporal correlations 55 56 57 58
-
In the former case, e.g., for a site-diluted contact process 55, activity tends to be restricted to favorable regions (lower than average dilution). In the present case, it is principally at the boundaries between active and inactive regions that, as implied by the Laplacian in Eq. (1), energy is transferred, and the effect is to move energy into the inactive region, thereby enhancing the further spread of activity. Indeed, the simulations reported below reveal none of the hallmarks of quenched disorder in the contact process, such as logarithmic time dependence in critical spreading, or generic power-law relaxation of temporal correlations 55 56 57 58.
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-
-
-
83
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3342997263
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Remarkably, the opposite identification is also possible in certain cases. See U. Alon, M.R. Evans, H. Hinrichsen, and D. Mukamel, Phys. Rev. Lett. 76, 2746 (1996).
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(1996)
Phys. Rev. Lett.
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Alon, U.1
Evans, M.R.2
Hinrichsen, H.3
Mukamel, D.4
-
84
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85036363351
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-
Note that in the case of quenched point disorder the “automaton” dynamics (i.e., when the local velocity can only take the values (Formula presented) is found to be in the same universality class as the continuous equation 28
-
Note that in the case of quenched point disorder the “automaton” dynamics (i.e., when the local velocity can only take the values (Formula presented) is found to be in the same universality class as the continuous equation 28.
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-
-
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85
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85036340376
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A.-L. Barabási and H.E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, 1995)
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A.-L. Barabási and H.E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, 1995).
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88
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85036340914
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It has been pointed out that the (Formula presented) function leads to an additional effective noise term 20, which could imply a different universality class for the automaton model and the continuous equation
-
It has been pointed out that the (Formula presented) function leads to an additional effective noise term 20, which could imply a different universality class for the automaton model and the continuous equation.
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-
-
-
89
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21844485611
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Note that the quenched disorder present in the LIM equations is mimicked in the RFT representation by the site-dependent non-Markovian term. The general equivalence between quenched noise and non-Markovian evolution was pointed out in the context of the so-called run-time-statistics; see M. Marsili, J. Stat. Phys. 77, 733 (1994).
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J. Stat. Phys.
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Marsili, M.1
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99
-
-
85036359100
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-
(unpublished)
-
R. Dickman, M. Alava, M.A. Muñoz, J. Peltola, A. Vespignani, and S. Zapperi (unpublished).
-
-
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Dickman, R.1
Alava, M.2
Muñoz, M.A.3
Peltola, J.4
Vespignani, A.5
Zapperi, S.6
-
100
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85036248242
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It is likely that, in fact, (Formula presented) is the exact result, as can be verified by using Priezzhev’s results 7. P. Grassberger (private communication)
-
It is likely that, in fact, (Formula presented) is the exact result, as can be verified by using Priezzhev’s results 7. P. Grassberger (private communication).
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-
-
-
101
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85036400730
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-
The strict linear relationship between (Formula presented) and (Formula presented) supports our elimination of (Formula presented) as an independent field, in the continuum description developed in Sec. III
-
The strict linear relationship between (Formula presented) and (Formula presented) supports our elimination of (Formula presented) as an independent field, in the continuum description developed in Sec. III.
-
-
-
-
103
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85036319722
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-
That (Formula presented) is insensitive to a change in the order of updating lends some support to the assertion that average properties are not strongly dependent on the kind of updating used for the BTW model
-
That (Formula presented) is insensitive to a change in the order of updating lends some support to the assertion that average properties are not strongly dependent on the kind of updating used for the BTW model.
-
-
-
-
105
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0002460727
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D.V. Ktitarev, S. Lubeck, P. Grassberger, and V.B. Priezzhev, Phys. Rev. E 61, 81 (2000).
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Phys. Rev. E
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4243410993
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M.A. Muñoz, R. Dickman, A. Vespignani, and Stefano Zapperi, Phys. Rev. E 59, 6175 (1999).
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Phys. Rev. E
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85036134590
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e-print cond-mat/9909347
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R. Dickman, e-print cond-mat/9909347.
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21344497779
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J.F.F. Mendes, R. Dickman, M. Henkel, and M.C. Marques, J. Phys. A 27, 3019 (1994).
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