Large-deviation properties of largest component for random graphs

Distributions of the size of the largest component, in particular the large-deviation tail, are studied numerically for two graph ensembles, for Erdös-Rényi random graphs with finite connectivity and for two-dimensional bond percolation. Probabilities as small as 10-180 are accessed using an artificial finite-temperature (Boltzmann) ensemble. The distributions for the Erdös-Rényi ensemble agree well with previously obtained analytical results. The results for the percolation problem, where no analytical results are available, are qualitatively similar, but the shapes of the distributions are somehow different and the finite-size corrections are sometimes much larger. Furthermore, for both problems, a first-order phase transition at low temperatures T within the artificial ensemble is found in the percolating regime, respectively