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

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

Offset: 7

Author

Hugo Pfoertner, Nov 08 2002

Keywords

• frac
• more
• nonn
• walk

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

Examples

			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)
		

References

• 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

• 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

Crossrefs

Cf. A077484, A077482, A001411.

Programs

Fortran

c See Hugo Pfoertner link.

Formula

P(n) = A077483(n) / ( 3^(n-1) * 2^A077484(n) )