  The Pairwise Einstein Model (PEM) of steps not only justifies the use of the
Generalized Wigner Distribution (GWD) for Terrace Width Distributions (TWDs),
it also predicts a specific form for the Step Position Distribution (SPD),
i.e., the probability density function for the fluctuations of a step about its
average position. The predicted form of the SPD is well approximated by a
Gaussian with a finite variance. However, the variance of the SPD measured from
either real surfaces or Monte Carlo simulations depends on $\Delta y$, the
length of step over which it is calculated, with the measured variance
diverging in the limit $\Delta y \to \infty$. As a result, a length scale
$L_{\rm W}$ can be defined as the value of $\Delta y$ at which the measured and
theoretical SPDs agree. Monte Carlo simulations of the terrace-step-kink model
indicate that $L_{\rm W} \approx 14.2 \xi_Q$, where $\xi_Q$ is the correlation
length in the direction parallel to the steps, independent of the strength of
the step-step repulsion. $L_{\rm W}$ can also be understood as the length over
which a {\em single} terrace must be sampled for the TWD to bear a "reasonable"
resemblence to the GWD.
