The assembly of molecular networks into structures such as random tilings and glasses has recently been demonstrated for a number of two-dimensional systems. These structures are dynamicallyarrested on experimental timescales so the critical regime in their formation is that of initial growth. Here we identify a transition from energetic to entropic stabilisation in the nucleation and growth of a molecular rhombus tiling. Calculations based on a lattice gas model show that clustering of topological defects and the formation of faceted boundaries followed by a slow relaxation to equilibrium occurs under conditions of energetic stabilisation. We also identify an entropicallystabilised regime in which the system grows directly into an equilibrium configuration without the need for further relaxation. Our results provide a methodology for identifying equilibrium and nonequilibrium randomness in the growth of molecular tilings, and we demonstrate that equilibrium spatial statistics are compatible with exponentially slow dynamical behaviour.The properties of two-dimensional supramolecular networks have been the focus of growing interest in recent years with most efforts directed towards the controlled introduction of translational order into such systems [1,2]. However there have been several recent observations of surface-bound supramolecular arrays which assemble into dynamically-arrested structures akin to glasses [3][4][5] which lack translational order. Such arrangements raise many interesting questions related to the growth of random systems [6,7]. In particular it is important to distinguish randomising effects which arise from kinetic effects, such as nucleation [8,9], from equilibrium disorder due to entropic terms in the free energy. Entropically-stabilised disorder may be regarded as intrinsic randomness, whereas kinetically-driven disorder is often determined by sample history and preparative conditions. In one recent study [3] a random molecular rhombus tiling was shown to have equilibrium (maximum entropy) spatial correlations, despite being frozen on an experimental timescale. In such a system the maximum entropy configuration must form, and be frozen in, during the initial growth, since the spatio-temporal fluctuations which normally facilitate the evolution of kinetically-trapped configurations to equilibrium are absent. However, it is not clear, a priori, that there is a set of local rules for molecular attachment which can lead to the direct growth of a 'perfect', i.e. maximum entropy, configuration.In this paper we address this question and show that equilibrium and non-equilibrium effects in the growth of a rhombus tiling [10-17] may be distinguished using tiletile correlations of arrays simulated using a lattice gas model [18]. Direct growth to a configuration with equilibrium statistics occurs when entropic terms dominate the free energy, while non-equilibrium effects result in faceted islands and clustering of topological defects.The parameters which control growth are the tile-tile interaction...