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Shape of a Self‐Avoiding Walk or Polymer Chain
1.M. E. Fisher and M. F. Sykes, Phys. Rev. 114, 45 (1959).
2.P. J. Flory, Principles of Polymer Chemistry (Cornell University Press, Ithaca, New York, 1953), Chap. 14.
3.S. F. Edwards, Proc. Phys. Soc. (London) 85, 613 (1965).
4.F. T. Wall and J. J. Erpenbeck, J. Chem. Phys. 30, 634 (1959), and earlier papers referred to therein.
5.M. E. Fisher and B. J. Hiley, J. Chem. Phys. 34, 1253 (1961).
6.J. Mazur, J. Chem. Phys. 38, 1292 (1963).
7.C. Domb, J. Chem. Phys. 38, 2957 (1963).
8.C. Domb, J. Gillis, and G. Wilmers, Proc. Phys. Soc. (London) 85, 625 (1965).
9.Actually Domb et al. fitted the distribution of one component of r, say x, to the simple form but this difference is not important. In three dimensions it corresponds simply to taking More recently J. Mazur, J. Res. Natl. Bur. Std. (U.S.) 69A, 355 (1965), has tried varying α and δ independently and found a range of fits to Monte Carlo estimates of With he found a good fit but he also presented evidence which suggested that was a better two‐parameter fit. He did not, however, test against the fits to the single‐component distributions.
10.See, for example, the discussion in M. E. Fisher, J. Math. Phys. 5, 944 (1964).
11.See, for example, W. Feller, Probability Theory and its Applications (John Wiley & Sons, Inc., New York, 1950), Vol. I, Chap. 7, especially Problem 14.
12.Edwards result (Ref. 3), which had was for a three‐dimensional walk which went on to infinity from the “end point” (n, r) and so is not strictly comparable to our
13.See, for example, C. Domb and M. E. Fisher, Proc. Cambridge Phil. Soc. 54, 48 (1958);
13.and H. N. V. Temperley, Phys. Rev. 103, 1 (1956).
14.A rigorous proof of an analog of this result for self‐avoiding walks terminating on a plane of perpendicular distance r from the origin, has been given by H. Kesten (private communication). However, rigorous justification of the remainder of the analysis, particularly the inversion of P̄(θ, r) to obtain Eq. (4.25), is still lacking and does not seem easy to supply.
15.L. Onsager, Phys. Rev. 65, 117 (1944).
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