  We propose stochastic $N$-component synchronization models
$(x_{1}(t),...,x_{N}(t))$, $x_{j}\in\mathbb{R}^{d}$, $t\in\mathbb{R}_{+}$,
whose dynamics is described by Levy processes and synchronizing jumps. We prove
that symmetric models reach synchronization in a stochastic sense: differences
between components $d_{kj}^{(N)}(t)=x_{k}(t)-x_{j}(t)$ have limits in
distribution as $t\rightarrow\infty$. We give conditions of existence of
natural (intrinsic) space scales for large synchronized systems, i.e., we are
looking for such sequences $\{b_{N}\}$ that distribution of
$d_{kj}^{(N)}(\infty)/b_{N}$ converges to some limit as $N\rightarrow\infty$.
It appears that such sequence exists if the Levy process enters a domain of
attraction of some stable law. For Markovian synchronization models based on
$\alpha$-stable Levy processes this results holds for any finite $N$ in the
precise form with $b_{N}=(N-1)^{1/\alpha}$. For non-Markovian models similar
results hold only in the asymptotic sense. The class of limiting laws includes
the Linnik distributions. We also discuss generalizations of these theorems to
the case of non-uniform matrix-based intrinsic scales. The central point of our
proofs is a representation of characteristic functions of $d_{kj}^{(N)}(t)$ via
probability distribution of a superposition of $N$ independent renewal
processes.
