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On the Number of Self‐Avoiding Walks. II
1.H. Kesten, J. Math. Phys. 4, 960 (1963).
2.For the walks in there is only Markov dependence (of order ) between the steps. Walks with Markov dependence have also been considered by E. W. Montroll, J. Chem. Phys. 18, 734 (1950). The author is indebted to the referee for this reference.
3.M. E. Fisher and M. F. Sykes, Phys. Rev. 114, 45 (1959).
4.For some more details see also the end of Sec. 2 and Ref. 15.
5.F. R. Gantmacher, Applications of the Theory of Matrices (Interscience Publishers, Inc., New York, 1959), Chap. III, Sec. 2.
6.J. M. Hammersley and D. J. A. Welsh, Quart. J. Math., Ser. 2 13, 108 (1962).
7.B. C. Rennie, Magy. Tud. Akad. Mat. Kut. Int. Kozlemen A6, 263 (1961).
8.J. M. Hammersley, Sankhya, A25, 269 (1963).
9.M. E. Fisher and D. S. Gaunt, Phys. Rev. 133A, 224 (1964).
10.L. V. Ahlfota, Complex Analysis (McGraw‐Hill Book Company, Inc., New York, 1953), Chap VI, Sec. 2.
11.J. M. Hammersley, Quart. J. Math., Ser. 2 12, 250 (1961).
12.J. M. Hammersley, Proc. Cambridge Phil. Soc. 57, 516 (1961).
13.We denote by “loop” the path where and not The number of points in this loop is which must be even.
14.The symbol means the probability of the event will be used for the conditional probability of E, given F. The simple random walk is a Markov chain on the integral points. It starts at and moves at each step from a point to one of its nearest neighbors, with probability for each neighbor. The individual steps are independent.
15.An analogous example is treated in detail in R. Bellman, Introduction to Matrix Analysis (McGraw‐Hill Book Company, Inc., New York, 1960), Chap. 16, Sec. 2–11.
16.G. Polya, Math. Ann. 84, 149 (1921).
17.The next‐to‐last member of (3.5) is correct only for but the last member can obviously be used for all
18.R. Courant, Differential and Integral Calculus (Interscience Publishers, Inc., New York), Vol. II, 2d ed., Appendix to Ch. IV, Sec. 3.3.
19.A. Renyi, Wahrscheinlichkeitsrechnung (VEB Deutscher Verlag der Wissenschaften, Berlin, 1962), p. 183.
20.We, also, use the inequality which is proved in exactly the same manner as the inequality in Ref. 11 [left‐hand inequality of (3)].
21.In two places we use [a] to denote the largest integer not exceeding a. In this meaning it occurs only as and as Even though we use square brackets in other meanings, the risk of confusion is small.
22.The cutoff point is chosen at because the estimate roughly takes its minimum for
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