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correlation function D ( n ) = 〈 S ( β $^{)}$( i ) S ( β $^{)}$( i + na⃗e ) 〉 . For both diffusive as well as RMT models we found no sig- nificant anisotropy, i.e. D was the same for ⃗e in the x or y direction. This is of course expected for the RMT model, but would deserve some more investigations in the diffusive case. The Fermi wavelength being equal to the lattice spacing close to half filling, one expects D ( n ) = δ$_{n}$$_{0}$ for the RMT model. This is confirmed in the inset of Fig. 3, where we additionally see that D ( n ) is also sharply peaked at n = 0 in the diffusive model, with some correlation persisting over few, ∼ 2-3 sites. This sharp decay of D ( n ) in the diffusive model suggests that our conjecture of a fast decay of the corre- lator 〈 T ( αβ ) P L T ( αβ ) P ′ L 〉 − 〈 T ( αβ ) P L 〉〈 T ( αβ ) P ′ L 〉 applies beyond RMT. This also deserves further investigations. 

Finally focusing on the RMT model, Fig. 3 confirms our main results, Eq. (14-15), by showing the scaling of the variance of the sample-integrated spin-dependent chemical potential in symmetric samples N$_{L}$ = N$_{R}$ = N as a function of N and the size M of the random matrix Hamiltonian. Small deviations from our analytical pre- dictions occur at small N for which our procedure of re- placing the denominator in Eq. (9) by its average squared when calculating var µ ( β ) P is no longer justified. All these results numerically confirm our scattering theory of spin polarizations. 

We have demonstrated how introducing an external probe allows to relate local current-induced spin polar- izations in a mesoscopic system to spin-resolved transmis- sion coefficients. Focusing on ballistic systems, we used random matrix theory to compute the statistics of these polarizations. We predicted that a ballistic quantum dot with large contacts to external transport leads – but still large dwell time – typically carries a spin polarization that exceeds those previously investigated by a paramet- rically large factor [see Eqs. (3–5)]. Perhaps most impor- tantly, we used this spin-probe model to calculate for the first time quantum correlations between local spin polar- izations and spin currents, beyond the classical Fick law. We showed that the reported large spin polarizations cor- relate only weakly with spin currents, thus they cannot be easily converted into spin currents – and vice-versa. As disappointing as this result may be, it is consistent with the universality of spin current fluctuations [16–19]. 

The reported accumulations are coherent in nature, and one might wonder how they disappear as the system is coupled to external sources of noise. In this respect it is interesting to note that for a multichannel probe, Eq. (16) becomes, for N, N$_{P}$ ≫ 1, 

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Enlarging the spin-probe damps the variance of the polar- izations in a similar way as a dephasing probe damps con- ductance fluctuations [34], with however a different para- metric dependence on N$_{P}$/N . Note that the probe gen- erating this damping is spin-conserving on time-average, as there is no spin current flowing through it. 

Beside decoherence effects, there are a number of in- teresting open problems, related for instance to spatial fluctuations of spin polarizations and our assumption of a sharp decay of correlations at k − 1 F . Finding out how the polarizations correlate to spin currents in diffusive systems, as well as in true ballistic systems treated be- yond RMT is also of interest. Ways to extract the large typical polarizations we reported in the nonlinear trans- port regime or by optical means as in Ref. [5] should also be explored. 

This work has been supported by the NSF under grant DMR-0706319. We acknowledge the hospitality of the Basel Center for Quantum Computing and Quantum Co- herence at the early stage of this project. We thank M. Duckheim and D. Loss for an introduction to magneto- electric effects and discussions of their letter, Ref. [23], and M. B¨ uttiker for discussions on probe physics. 

## V. CONCLUSIONS 

## Acknowledgments 

