In the paper "Balls and bins a simple and tight analysis" by Raab and Steger, available here strong upper and lower bounds are proved about the number $M$ of balls in a maximally loaded bin when $m$ balls are independently and uniformly thrown into $n$ bins.

For example, if $$n \log n \ll m \leq n \cdot polylog (n) $$ then $$Pr[M >k_{\alpha}]= o(1),\qquad~~ if ~~\alpha> 1$$ and $$Pr[M >k_{\alpha}]=1−o(1),~\qquad~ if ~~0< \alpha<1,$$ where $$k_{\alpha}=\frac{m}{n}+\alpha \sqrt{ 2\frac{m}{n}~ \log n}.$$

I am interested in estimates of the variance of $M.$

Also, I know that $M$'s expectation is exactly of order

$$\log n/\log \log n$$ if $m=n,$ but not in general.