TOPICS
Search

s-Run


Let v be a n-vector whose entries are each 1 (with probability p) or 0 (with probability q=1-p). An s-run is an isolated group of s consecutive 1s. Ignoring the boundaries, the total number of runs R_n satisfies

K_n=(<R_n>)/n
(1)
=(1-p)^2sum_(s=1)^(n)p^s
(2)
=p(1-p)(1-p^n),
(3)

so

K(p)=lim_(n->infty)K_n
(4)
=p(1-p),
(5)

which is called the mean run count per site or mean run density in percolation theory.


See also

Percolation Theory, s-Cluster

Explore with Wolfram|Alpha

References

Finch, S. R. "Percolation Cluster Density Constants." §5.18 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 371-378, 2003.

Referenced on Wolfram|Alpha

s-Run

Cite this as:

Weisstein, Eric W. "s-Run." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/s-Run.html

Subject classifications