Large-deviation properties of largest component for random graphs

Distributions of the size of the largest component, in particular the large-deviationtail, are studied numerically for two graph ensembles, for Erdös-Rényi random graphs withfinite connectivity and for two-dimensional bond percolation. Probabilities as small as10^-180 are accessed using an artificial finite-temperature (Boltzmann)ensemble. The distributions for the Erdös-Rényi ensemble agree well with previouslyobtained analytical results. The results for the percolation problem, where no analyticalresults are available, are qualitatively similar, but the shapes of the distributions aresomehow different and the finite-size corrections are sometimes much larger. Furthermore,for both problems, a first-order phase transition at low temperatures Twithin the artificial ensemble is found in the percolating regime, respectively.