We denote by $||.||_1$ the trace class norm. on $M_n$.Let $(r_{ij})_{1 \leq i,j\leq n}$ be a family of independent identically distributed random variables which take the values $-1$ and $1$ with equal probability.
Let $e_{ij}$ be the standard unit matrix of $M_n$. We define the random matrix $$ A=\sum_{i,j=1}^n r_{ij}e_{ij}. $$ I am interested by lower bounds of the quantities $$ P\big(||A||_1<\alpha\mathbb{E}||A||_1\big)\ \text{ and } \ P\big(||A||_1>\beta\mathbb{E}||A||_1\big) $$
where $\alpha, \beta>0$.
I would like obtain the existence of random matrices with $||.||_1$ far from the expectation (as far away as possible).
Any comment is welcome.
Remark 1: it seems to me that Szarek proved that $$ \mathbb{E}(||A||_1)=O(n^{3/2}). $$
Remark 2: If $B$ is a matrix of $M_n$ such each entry of $B$ is 1 or -1 then $$ ||B||_1\leq K_n $$ where $K_n$ is a constant depending on the dimension $n$.