Given a $n\times n$ symmetric random matrix such that

  1. all diagonal elements are all fixed as $0$.
  2. randomly select $k$ distinct cells in the upper triangle (excluding the diagonal), and then fix them as $1$. The elements in the lower triangle are set accordingly. Of course $k\le \frac{n^2-n}{2}$.
  3. all other elements are independent uniform random variables over $[0,1]$.

The question is

Any known result for the distribution of the minimum/maximum row sum?

It is easier to just look at one row. Let $c$ be the number of chosen cells to be fixed as $1$, then its distribution is $\frac{{\left( {\begin{array}{*{20}{c}} {n - 1} \\ c \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\frac{{{n^2} - n}}{2} - (n - 1)} \\ {k - c} \end{array}} \right)}}{{\left( {\begin{array}{*{20}{c}} {\frac{{{n^2} - n}}{2}} \\ k \end{array}} \right)}}$, which equals the probability of $c$ chosen cells in the first row. The sum of the remaining $n-1-c$ cells are i.i.d. uniform, and their sum is Irwin–Hall distribution.

However, I have trouble to find the distribution for min/max row sum, as the row sums are not even independent.

A related question is Bound for largest eigenvalue of symmetric matrices of uniform random variables over $[0,1]$ and fixed $1$s along diagonal and scattered $1$s.


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