  We compute analytically the probability of large fluctuations to the left of
the mean of the largest eigenvalue in the Wishart (Laguerre) ensemble of
positive definite random matrices. We show that the probability that all the
eigenvalues of a (N x N) Wishart matrix W=X^T X (where X is a rectangular M x N
matrix with independent Gaussian entries) are smaller than the mean value
<\lambda>=N/c decreases for large N as $\sim \exp[-\frac{\beta}{2}N^2
\Phi_{-}(\frac{2}{\sqrt{c}}+1;c)]$, where \beta=1,2 correspond respectively to
real and complex Wishart matrices, c=N/M < 1 and \Phi_{-}(x;c) is a large
deviation function that we compute explicitly. The result for the Anti-Wishart
case (M < N) simply follows by exchanging M and N. We also analytically
determine the average spectral density of an ensemble of constrained Wishart
matrices whose eigenvalues are forced to be smaller than a fixed barrier. The
numerical simulations are in excellent agreement with the analytical
predictions.
