We consider the ε-approximation by n-term partial sums of the Karhunen- Lo`eve expansion to d-parametric random fields of tensor product-type in the average case setting. We investigate the behavior, as d → ∞, of the informa- tion complexity of approximation with error not exceeding a given level ε. It was recently shown that for this problem one observes the curse of dimen- sionality (intractability) phenomenon. We aim to give the exact asymptotic expression for the information complexity.