Let $p \in ]0,1[$. We start from the zero matrix of $\mathcal{M}_n(\mathbb{R})$, denoted $M_0$. For all $k \in \mathbb{N}$, we construct the matrix $M_{k+1}$ from the matrix $M_k$ by scanning through the matrix in one pass and changing each zero coefficient to 1 with probability $p$, independently. We denote $q = 1-p$ and $m = n^2$. Let $i$ and $j$ be in $\{1, \ldots, n\}$. The smallest integer $k \geqslant 1$ such that the coefficient at row $i$, column $j$ of $M_k$ equals 1 is denoted $T_{i,j}$. Give the distribution of $T_{i,j}$.
Let $p \in ]0,1[$. We start from the zero matrix of $\mathcal{M}_n(\mathbb{R})$, denoted $M_0$. For all $k \in \mathbb{N}$, we construct the matrix $M_{k+1}$ from the matrix $M_k$ by scanning through the matrix in one pass and changing each zero coefficient to 1 with probability $p$, independently. We denote $q = 1-p$ and $m = n^2$.
Let $i$ and $j$ be in $\{1, \ldots, n\}$. The smallest integer $k \geqslant 1$ such that the coefficient at row $i$, column $j$ of $M_k$ equals 1 is denoted $T_{i,j}$. Give the distribution of $T_{i,j}$.