

A077483


Probability P(n) of the occurrence of a 2D selftrapping walk of length n: Numerator.


5



2, 5, 31, 173, 1521, 1056, 16709, 184183, 1370009, 474809, 13478513, 150399317, 1034714947, 2897704261
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OFFSET

7,1


COMMENTS

A comparison of the exact probabilities with simulation results obtained for 1.2*10^10 random walks is given under "Results of simulation, comparison with exact probabilities" in the first link. The behavior of P(n) for larger values of n is illustrated in "Probability density for the number of steps before trapping occurs" at the same location. P(n) has a maximum for n=31 (P(31)~=1/85.01) and drops exponentially for large n (P(800)~=1/10^9). The average walk length determined by the numerical simulation is sum n=7..infinity (n*P(n))=70.7598+0.001


REFERENCES

See under A001411
Alexander Renner: Self avoiding walks and lattice polymers. Diplomarbeit University of Vienna, December 1994
More references are given in the sci.math NG posting in the second link


LINKS

Table of n, a(n) for n=7..20.
Hugo Pfoertner, Results for the 2D SelfTrapping Random Walk
Hugo Pfoertner, Selftrapping random walks on square lattice in 2D (cubic in 3D).Posting in NG sci.math dated March 4, 2002


FORMULA

P(n) = a077483(n) / ( 3^(n1) * 2^a077484(n) )


EXAMPLE

A077483(10)=173 and A077484(10)=1 because there are 4 different probabilities for the 50 (=2*A077482(10)) walks: 4 walks with probability p1=1/6561, 14 walks with p2=1/8748, 22 walks with p3=1/13122, 10 walks with p4=1/19683. The sum of all probabilities is P(10) = 4*p1+14*p2+22*p3+10*p4 = (12*4+9*14+6*22+4*10)/78732 = 346/78732 = 173 / (3^9 * 2^1)


PROG

FORTRAN program provided at first link


CROSSREFS

Cf. A077484, A077482, A001411.
Sequence in context: A107389 A261750 A189559 * A119242 A068145 A032112
Adjacent sequences: A077480 A077481 A077482 * A077484 A077485 A077486


KEYWORD

frac,more,nonn,walk


AUTHOR

Hugo Pfoertner, Nov 08 2002


STATUS

approved



