Au-substrate interaction was calculated for a number of different positions of an Au atom relative to the substrate surface. For each position the Au-substrate interaction was mapped out by holding a Au atom fixed at different heights above the surface while the carbon atoms at and near the surface were allowed to fully relax. As a reference point a Au atom at a distance of 9.5 Å from the surface was used (Sec. \[SubstrateDFT\]). The resulting Au-substrate potentials were averaged and fitted to a modified Lennard-Jones potential and used as input in MC simulations described in the next section.

Theory: Monte-Carlo (MC) simulations {#sectionDefMC}
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In order to develop a microscopic understanding of the dynamics of the Au clusters on the substrate we simulate the system with the canonical Monte-Carlo method. A standard Metropolis algorithm is employed [@AT89; @WDS+01; @Wern05a] with an update after each random displacement of an Au atom within an interval $[0,d_{\rm max}]$ in all spatial dimensions. $d_{\rm max}$ is set to yield an MC acceptance rate of 50 to 60 %. The resulting temperature dependence is roughly $d_{\rm max} \propto \sqrt{T}$. The boundary conditions are imposed by a hard wall cube with linear dimension $L_x$, $L_y$, 