In the paper "Balls and bins a simple and tight analysis" by Raab and Steger, available [here](http://www.dblab.ntua.gr/~gtsat/collection/scheduling/Balls%20into%20Bins%20A%20Simple%20and%20Tight%20Anal.pdf) 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.



Variance of load in maximally loaded bin, if $m$ balls are thrown into $n$ bins
Theorem 4.4 in Devroye's book *On Bucket Algorithms* answers this question. His $n,m$ are opposite to those in the question and uniformity is not assumed. Subject to some technical conditions on a nonlinear, nonnegative, convex fumction $g(x)$ on $[0,\infty)$ which are satisfied here, with $g(x)=x^2,$ 
his result can be written as
$$
\mathbb{E}(g(M))=(1+o(1))g(\log n/\log\log n)
$$
which yields by subtraction
$$
\textrm{Var}(M)=f(n,m) \times \left(\frac{\log n}{\log \log n}\right)^2
$$
where
$$
f(n,m)\sim \frac{n e^{-u \log\left(un/emq\right)}e^{-mq/n}}{\log\left(un/emq\right)}
$$
where $u=(1+\varepsilon) \log n/\log\log n$, and $q=n\max\_i \{p\_i\}=1$ here which yields
$$
f(n,m)\sim \frac{n e^{-u \log\left(un/em\right)}e^{-m/n}}{\log\left(un/em\right)}.
$$


