Capture-Zone Areas \& the Wigner Distribution: New Case of Universal Scaling of Spacings in Fluctuating Systems

When investigating scaling of island sizes during growth in $d$ dimensions, one should consider the distribution of the areas of proximity cells around nucleation centers, i.e. capture zones (CZ). Using data from kinematic Monte Carlo studies,\footnote{ Mulheran et al., PRB {\bf 53} ('96) 10261, {\bf 54} ('96) 11681; EPL {\bf 49} ('00) 617, {\bf 65} (’04) 379. Amar, Family, et al., PRL {\bf 74} ('95) 2066; PRB {\bf 64} (’01) 205404. Evans, Bartelt, et al. PRB {\bf 66} (’02) 235410; SSR {\bf 61} ('06) 1.} we find that the CZ distributions in both $d$ = 1 and $d$ = 2 are well described by the generalized Wigner distribution (GWD) from random-matrix theory: $P_\varrho(s)=as^\varrho\exp(-bs^2)$. $P_\varrho(s)$ accounts for a broad range of fluctuation phenomena, inc.\ the terrace-width distribution (TWD) on vicinal surfaces. For CZ distributions, we find $\varrho = i + d/2$, where $i$ is the critical nucleus size. We present a phenomenological justification by constructing a Langevin equation similar to that used in accounting for the equilibration of TWDs.\footnote{A. Pimpinelli, H. Gebremariam, \& T.L. Einstein, PRL 95 ('05) 246101} We discuss implications for processing and analysis of experimental data.