  We prove that the number of iterations required to solve a random positive
definite linear system with the conjugate gradient algorithm is almost
deterministic for large matrices. We treat the case of Wishart matrices $W =
XX^*$ where $X$ is $n \times m$ and $n/m \sim d$ for $0 < d < 1$. Precisely, we
prove that for most choices of error tolerance, as the matrix increases in
size, the probability that the iteration count deviates from an explicit
deterministic value tends to zero. In addition, for a fixed iteration count, we
show that the norm of the error vector and the norm of the residual converge
exponentially fast in probability, converge in mean and converge almost surely.
