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

Abstract

We prove a moderate deviation principle for subgraph count statistics of Erdős-Rényi 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 log-Laplace 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.Here (e κ 1 , . . . , e κ k ) denotes the graph with edges e κ 1 , . . . , e κ k present and A ∼ G denotes the fact that the subgraph A of the complete graph is isomorphic to G. We assume G to be a

Moderate deviations of subgraph counts in the Erdős-Rényi random graphs 𝐺(𝑛,𝑚) and 𝐺(𝑛,𝑝)

“…already appears in the article of Döring and Eichelsbacher [9]. The difference between the results is the order of magnitude of the error term.…”

“…The difference between the results is the order of magnitude of the error term. In the range δ n ≫ n −1/2 √ log n we have an error term of the form o(δ 2 n n) in the exponent 2 On the other hand, the result of Döring and Eichelsbacher [9] holds whenever…”

Moderate Deviations via Cumulants

“…The stated moderate deviation principle in Theorem 2.3 is on one hand valid for more probabilities p(n) than in [11, Theorem 1.1]. But on the other hand the scaling β n := a n √ VW has a smaller range in comparison to [11]: Using [29, 2nd section, page 5]) the scaling in Theorem 2.3 is equal to…”

“…Condition (2.13) on p(n) assures that (a n ) n grows to infinity. Moderate deviations for the subgraph count statistic of Erdős-Rényi random graphs are already considered in [11] studying the log-Laplace transform via martingale differences and using the Gärtner-Ellis Theorem. The stated moderate deviation principle in Theorem 2.3 is on one hand valid for more probabilities p(n) than in [11, Theorem 1.1].…”

Edge Fluctuations of Eigenvalues of Wigner Matrices

“…This moderate deviations result does not have yet a fully universal version for Wigner matrices. It has been generalised to Gaussian divisible matrices with a deterministic self-adjoint matrix added with converging empirical measure [7] and to Bernoulli matrices [9]. Recently we proved in [10] a MDP for the number of eigenvalues of a GUE matrix in an interval.…”