Moderate Deviations in a Random Graph and for the Spectrum of Bernoulli Random Matrices

We prove a moderate deviation principle for subgraph count statistics of ErdősRényi random graphs. This is equivalent in showing a moderate deviation principle for the trace of a power of a Bernoulli random matrix. It is done via an estimation of the logLaplace transform and the Gärtner-Ellis theorem. We obtain upper bounds on the upper tail probabilities of the number of occurrences of small subgraphs. The method of proof is used to show supplemental moderate deviation principles for a class of symmetric statistics, including non-degenerate U-statistics with independent or Markovian entries.