s-Run

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Let ${\bf v}$ 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

$$K_n={\left\langle{R_n}\right\rangle{}\over n}=(1-p)^2\sum_{s=1}^n p^s=p(1-p)(1-p^n),$$

so

$$K(p)\equiv \lim_{n\to\infty} K_n = p(1-p),$$

which is called the Mean Run Count Per Site or Mean Run Density in Percolation Theory.

See also Percolation Theory, s-Cluster

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/rndprc/rndprc.html