Download Cumulants for random matrices and their asymptotic behavior .

Cumulants for random matrices and their asymptotic behavior.
(M. Casalis, M. Capitaine)
The asymptotic freeness between two independent sets of matrices can be proved
from the convergence of the mixte generalized moments E(rπ (B1 X1 , ..., Bn Xn )) =
Q
Q
Q
E( ki=1 T r( j∈Ci Bj Xj )) (where π = ki=1 Ci is a permutation on {1, 2, ..., n} decomposed into cycles Ci ) towards the corresponding mixte moment φπ (b1 x1 , ..., bn xn ) =
Qk
Q
i=1 φ( j∈Ci bj xj ) between two free sets of non commutative variables.
We prove here that under the hypothesis that the matrices are complex and unitary
invariant, such moments looked as a function on the symmetric group Sn satisfy
a convolution relation between the generalized moments of the Bj and some function cX1 ,...,Xn on Sn . Matricial cumulants are then defined from cX1 ,...,Xn . We prove
first that they linearize the convolution of probability measures and then that they
converge towards the corresponding free cumulants, while the convolution on Sn
defining the mixte moments becomes the convolution on non crossing partitions existing for free mixte moments.
The same development can be realized for real matrices which are invariant under
the orthogonal group. There we show that the previous convolution does no more
occur on Sn but on S2n .
1