Suppose $A$ is a random $n \times n$ matrix with integer entries in respect to the orthonormal basis $(e_i)_{i=1}^{n} \subset \mathbb{R}^n$. Define $\{v_t\}_{t=1}^{\infty}$ as a sequence of random vectors in $\mathbb{R}^n$ satisfying the following relationships: $$P(v_1 = e_1) = 1$$ $$v_{t+1} = \sum_{i=1}^n \sum_{j = 1}^{\langle v_t, e_i \rangle} A_{ijt}\,e_i$$ Here $A_{ijt}$ are i.i.d «copies» of $A$.This definition is correct as $\forall i \leq n P(\langle v_t, e_i \rangle \in \mathbb{Z}) = 1$. What conditions on $A$ result in $\lim\limits_{n \to \infty} P(v_n = 0) = 1$?
If $n = 1$, then $\{v_t\}_{t=1}^{\infty}$ form a Galton-Watson branching process (with $A$ becoming a scalar random variable, that can be interpreted as the number of direct successors).In that case, it is known, that $\lim\limits_{n \to \infty} P(v_n = 0) = 1$ iff $EA \leq 1$. However, I do not know how this statement can be generalised for an arbitrary natural $n$.
According to "A first course in Stochastic Processes" by Samuel Karlin and Howard M. Taylor, this process is called "Multi-Type Branching Process". The extinction criterion for it is: