Concentration of measure for the number of isolated vertices in the Erd\H{o}s-R\'{e}nyi random graph by size bias couplings

A concentration of measure result is proved for the number of isolated vertices $Y$ in the Erd\H{o}s-R'{e}nyi random graph model on $n$ edges with edge probability $p$. When $\mu$ and ${\sigma }^2$ denote the mean and variance of $Y$ respectively, $P((Y-\mu )\mathrm{/}\sigma \ge t)$ admits a bound of the form $e^{-k{t}^2}$ for some constant positive $k$ under the assumption $p\in (0,1)$ and $np\to c\in (0,\mathrm{\infty })$ as $n\to \mathrm{\infty }$. The left tail inequality P(\frac{Y-\mu}{\sigma}\le -t)&\le& \exp(-\frac{t^2\sigma^2}{4\mu}) holds for all $n\in 2,3,\mathrm{.}\mathrm{.}\mathrm{.},p\in (0,1)$ and $t\ge 0$. The results are shown by coupling $Y$ to a random variable $Y^s$ having the $Y$-size biased distribution, that is, the distribution characterized by $E[Yf(Y)]=\mu E[f(Y^s)]$ for all functions $f$ for which these expectations exist.