Random walks and the symplectic representation of the braid group

We consider the symplectic representation ${\rho }_n$ of a braid group $B(n)$ in $Sp(2l,\mathbb{Z})$, for $l=\left[{\displaystyle \frac{n-1}{2}}\right]$. If $P$ is a polynomial on the $4l^2$ 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 $B(n)$. We derive that the $n$-braids $\beta$ which close into a loop $\widehat{\beta }$ with $0<\mathrm{\mid }det(\widehat{\beta })\mathrm{\mid }\le C$ for some constant $C$ form a transient set. And given a prime number $p$, we show that the probability for a given braid to close in a $p$-colorable loop is greater than ${\displaystyle \frac{1}{p}}$. 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 $1\le i,j\le 2$. \ As an example of such braids, we investigate the signature of the Lissajous toric knots $3$-braids.