  Let W be a Wishart random matrix of size d^2 times d^2, considered as a block
matrix with d times d blocks. Let Y be the matrix obtained by transposing each
block of W. We prove that the empirical eigenvalue distribution of Y approaches
a non-centered semicircular distribution when d tends to infinity. We also show
the convergence of extreme eigenvalues towards the edge of the expected
spectrum. The proofs are based on the moments method.
  This matrix model is relevant to Quantum Information Theory and corresponds
to the partial transposition of a random induced state. A natural question is:
"When does a random state have a positive partial transpose (PPT)?". We answer
this question and exhibit a strong threshold when the parameter from the
Wishart distribution equals 4. When d gets large, a random state on C^d tensor
C^d obtained after partial tracing a random pure state over some ancilla of
dimension alpha.d^2 is typically PPT when alpha>4 and typically non-PPT when
alpha<4.
