Terrace Width Distributions in the Limit $\tilde{\beta}_B / \tilde{\beta}_A \to \infty$: Numerical Transfer Matrix Results

With a few physical and mathematical simplifications, the Terrace Width Distributions (TWDs) for a stepped crystal surface with typical step interactions have been shown to be Generalized Wigner Distributions (GWDs).\footnote{H.~L.\ Richards and T.~L.Einstein, \textit{Phys.\ Rev.\ E} {\bf 76}, 016124 (2005).} This is true even when steps have different stiffnesses ($\tilde{\beta}_A$ and $\tilde{\beta}_B$) that alternate, as has been confirmed by Monte Carlo simulations.\footnote{J.~A.\ Yancey, H.~L.\ Richards, and T.~L.Einstein, \textit{Surf.\ Sci.\/} {\bf 598}, 78--87 (2005).} Monte Carlo simulations have three serious drawbacks for studying very unequal stiffnesses: (1) the simulated steps have only finite length, which may be close to or smaller than the correlation length; (2) the time required to equilibrate may be prohibitively long; and (3) statistical uncertainties are unavoidable. Additionally, the simulations in Ref.~2 were problematic, since $\tilde{\beta}_A$ approached zero as $\tilde{\beta}_B$ became large. This work avoids all those problems by finding TWDs from numerical transfer matrices with $\tilde{\beta}_A$ held constant as $\tilde{\beta}_B \to \infty$. This is a necessary step before the GWD can be analyzed as an ensemble average of Gruber-Mullins TWDs.