quantum chaos: in the classical evolution of $X$ and in the quantum dynamics governed by $H^{eff}$.

Horizon of coherence of the reduced density matrix
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Let $|\psi {\rangle \hspace{-0.2em} \rangle}$ be the state of the fermionic string solution of eq. \[equSchro\]. We consider the reduced density matrix: $$\rho = \tr_{\mathbb C^N} |\psi {\rangle \hspace{-0.2em} \rangle}{\langle \hspace{-0.2em} \langle}\psi|$$ corresponding to the mixed state of the spin of the string. This spin is subject to the entanglement with the string attachment degree of freedom, to the classical chaotic fluctuations of $X$ and $x$, and to the quantum chaos induced by $H^{eff}$ after the thermalization. In previous works [@Viennot2; @Aubourg; @Aubourg2], we have shown that a quantum system submitted to a chaotic flow presents an horizon of coherence, an initial duration where the quantum system is preserved from the decoherence phenomenon induced by the chaos. This horizon of coherence is a quantum version of the horizon of predictability of the classical chaotic flows (see discussion in [@Viennot2]). In order to study this point for the spin of the fermionic string, we consider the coherence $|\langle \uparrow|\overline \rho|\downarrow \rangle|$ of the averaged density matrix $\overline \rho$ (the average being realized on the initial conditions 