A077483 Probability P(n) of the occurrence of a 2D self-trapping walk of length n: Numerator.

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

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 Self-Trapping Random Walk

Hugo Pfoertner, Self-trapping random walks on square lattice in 2-D (cubic in 3-D).Posting in NG sci.math dated March 4, 2002

Formula

P(n) = a077483(n) / ( 3^(n-1) * 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) c 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