Abstract
We study the two-dimensional first passage problem in which bonds have zero and unit passage times with probabilityp and 1−p, respectively. We prove that as the zero-time bonds approach the percolation thresholdp c, the first passage time exhibits the same critical behavior as the correlation function of the underlying percolation problem. In particular, if the correlation length obeysξ(p) ∼¦p−p c¦−v, then the first passage time constant satisfiesμ(p)∼¦p−p c¦v. At pc, where it has been asserted that the first passage time from 0 tox scales as ¦x¦ to a power ψ with 0<ψ<1, we show that the passage times grow like log ¦x¦, i.e., the fluid spreads exponentially rapidly.
Similar content being viewed by others
References
J. M. Hammersley and D. J. A. Welsh, inBernoulli, Bayes, Laplace Anniversary Volume, J. Neyman and L. M. LeCam, eds. (Springer-Verlag, 1965), p. 61.
R. T. Smythe and J. C. Wierman,First Passage Percolation on the Square Lattice (Lecture Notes in Mathematics, Vol. 671, Springer-Verlag, 1978).
H. Kesten, Aspects of first passage percolation, Cornell preprint (1985).
A. L. Ritzenberg and R. C. Cohen,Phys. Rev. B 30:4038 (1984).
A. R. Kerstein,Phys. Rev. B 31:321 (1985).
A. R. Kerstein,Phys. Rev. B 30:2980 (1984).
R. Chandler, J. Koplick, K. Lerman, and J. F. Willemsen,J. Fluid Mech. 119:249 (1982).
D. Wilkinson and J. F. Willemsen,J. Phys. A: Math. Gen. 16:3365 (1983).
J. T. Chayes, L. Chayes, and C. M. Newman,Commun. Math. Phys. 101:383 (1985).
J. F. C. Kingman,J. R. Stat. Soc. B 30:499 (1968).
J. T. Cox and R. Durrett, in preparation.
H. Kesten,Adv. Appl. Prob. 12:848 (1980).
H. Kesten, Surfaces with minimal random weights and maximal flows: A higher dimensional version of first passage percolation, Cornell preprint (1985).
H. Kesten, First-passage percolation and a higher dimensional generalization, Cornell preprint (1985).
J. T. Cox and R. Durrett,Ann. Prob. 9:583 (1981).
J. C. Wierman,Adv. Appl. Prob. 9:283 (1977).
J. T. Chayes and L. Chayes, Percolation and random media, Harvard preprint (1985); to appear inCritical Phenomena, Random Systems and Gauge Theories, Les Houches Session XLII1 1984, K. Osterwalder and R. Stora, eds. (Elsevier).
S. R. Broadbent and J. M. Hammersley,Proc. Camb. Phil. Soc. 53:629 (1957).
J. M. Hammersley,Ann. Math. Stat. 28:790 (1957).
H. Kesten,Commun. Math. Phys. 74:41 (1980).
M. Aizenman and D. Barsky, in preparation.
G. R. Grimmett,J. Phys. A: Math. Gen. 16:599 (1983).
J. T. Chayes, L. Chayes, and J. Fröhlich,Commun. Math. Phys. 100:399 (1985).
B. Nyugen, Thesis, UCLA (1985).
H. Kesten, Scaling relations for 2D percolation, IMA preprint (1986).
L. Russo,Z. Wahrsch. verw. Geb. 43:39 (1978).
P. D. Seymour and D. J. A. Welsh,Ann. Discrete Math. 3:227 (1978).
H. Kesten,Percolation Theory for Mathematicians (Birkhauser, 1982).
T. E. Harris,Proc. Camb. Phil. Soc. 56:13 (1960).
R. Lenormand and S. Bories,C. R. Acad. Sci. Paris B 291:297 (1980).
J. van den Berg and H. Kesten, Inequalities with applications to percolation and reliability, preprint (1984); to appear inAdv. Appl. Prob.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chayes, J.T., Chayes, L. & Durrett, R. Critical behavior of the two-dimensional first passage time. J Stat Phys 45, 933–951 (1986). https://doi.org/10.1007/BF01020583
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01020583