# Self-trapping 2D random walks

Pfoertner, Hugo Hugo.Pfoertner at muc.mtu.de
Fri Nov 8 16:14:54 CET 2002

```Hi Seqfans,

in OEIS)

%I A077482
%S A077482 1,2,11,25,95,228,752,1860,5741,14477,42939,109758,317147,818229
%N A077482 Number of self-avoiding walks trapped after n steps.
%C A077482 Only 1/8 of all possible walks is counted by selecting the first
step in +x direction and requiring the firs\
t step changing y to be positive.
%D A077482 See references given for A001411
%H A077482 Hugo Pfoertner, <a
href="http://www.randomwalk.de/stw2d.html">Results for the 2D Self-Trapping
Random Walk</\
a>
%e A077482 a(7)=1 because there is only 1 self-trapping walk with 7 steps:
(0,0)(1,0)(1,1)(1,2)(0,2)(-1,2)(-1,1)(0,1) a\
(8)=2 because there are 2 self-trapping walks with 8 steps:
(0,0)(1,0)(2,0)(2,1)(2,2)(1,2)(0,2)(0,1)(1,1) (0,0)(1,0)(\
1,1)(2,1)(3,1)(3,0)(3,-1)(2,-1)(2,0)
%o A077482 Fortran program provided at given link
%Y A077482 Cf. A046661, A001411.
%K A077482 more,nice,nonn,new
%O A077482 7,2
%A A077482 Hugo Pfoertner (all at abouthugo.de), Nov 07 2002

Submitted:

> %I A077483
> %S A077483 2 5 31 173 1521 1056 16709 184183 1370009 474809 13478513
> 150399317 1034714947 2897704261
> %N A077483 Probability P(n) of the occurrence of a 2D self-trapping walk
of length n: Numerator
> %C A077483 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
> 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
> %D A077483 See under A001411
> %D A077483 Alexander Renner: Self avoiding walks and lattice polymers.
Diplomarbeit University of Vienna, December 1994
> %D A077483 More references are given in the sci.math NG posting in the
> %H A077483 Hugo Pfoertner, <a
href="http://www.randomwalk.de/stw2d.html">Results for the 2D
Self-Trapping Random Walk</a>
> %H A077483 Hugo Pfoertner, <a
href="http://mathforum.org/discuss/sci.math/t/394788">Self-trapping
random walks on square lattice in 2-D (cubic in 3-D).Posting in NG sci.math
dated March 4, 2002</a>
> %F A077483 P(n) = a077483(n) / ( 3^(n-1) * 2^a077484(n) )
> %e A077483 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)
> %o A077483 Fortran program provided at first link
> %Y A077483 Cf. A077484, A077482, A001411
> %O A077483 7
> %K A077483 ,frac,more,nice,nonn,
> %A A077483 Hugo Pfoertner (all at abouthugo.de), Nov 08 2002

> %I A077484
> %S A077484 0 0 0 1 2 0 2 4 5 2 5 7 8 8
> %N A077484 Probability P(n) of the occurrence of a 2D self-trapping walk
of length n: Exponent of 2 in the denominator
> %C A077484 This sequence is the exponent of 2 in the denominator for P(n).
> %D A077484 See under A077483
> %F A077484 P(n) = A077483(n) / ( 3^(n-1) * 2^A077484(n) )
> %e A077484 See under A077483
> %Y A077484 Cf. A077483
> %O A077484 7
> %K A077484 ,frac,more,nice,nonn,
> %A A077484 Hugo Pfoertner (all at abouthugo.de), Nov 08 2002

Thanks,
Hugo

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