  We consider the symplectic representation $\rho_n$ of a braid group $B(n)$ in
$Sp(2l,\mathbb{Z})$, for $l=\Big[\dfrac{n-1}{2}\Big]$. If $P$ is a $4l^2$
polynomial on the coefficients of the matrices in $Sp(2l,\mathbb{Z})$, we show
that the set $\{\beta\in B(n): P(\rho_n(\beta))=0\}$ is transient for non
degenerate random walks on $B(n)$.
  If $n$ is odd, we derive that the $n$-braids $\beta$ verifying
$|\Delta_{\hat{\beta}}(-1)|\leq C$ for some constant $C$ form a transient set:
here $\Delta_{\hat{\beta}}$ denotes the Alexander polynomial of the closure of
$\beta$.
  We also derive that for a random $3$-braid, the quasipositive links
$(\beta\sigma_i\beta^{-1}\sigma_j)^p$ have zero signature for every integer $p$
and $1\leq i,j\leq 2$. \\ As an example of such braids, we investigate the
signature of the Lissajous toric knots with $3$ strands.
