Concentration of measure for the number of isolated vertices in the Erdős–Rényi random graph by size bias couplings

A concentration of measure result is proved for the number of isolated vertices YY in the Erdős–Rényi random graph model on nn edges with edge probability pp. When μμ and σ2σ2 denote the mean and variance of YY respectively, P((Y−μ)/σ≥t)P((Y−μ)/σ≥t) admits a bound of the form e−kt2e−kt2 for some constant positive kk under the assumption p∈(0,1)p∈(0,1) and np→c∈(0,∞)np→c∈(0,∞) as n→∞n→∞. The left tail inequality P(Y−μσ≤−t)≤exp(−t2σ24μ) holds for all n∈{2,3,…},p∈(0,1)n∈{2,3,…},p∈(0,1) and t≥0t≥0. The results are shown by coupling YY to a random variable YsYs having the YY-size biased distribution, that is, the distribution characterized by E[Yf(Y)]=μE[f(Ys)]E[Yf(Y)]=μE[f(Ys)] for all functions ff for which these expectations exist.