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 × N ) Wishart matrix W = X T X (where X is a rectangular M ×N matrix with independent Gaussian entries) are smaller than the mean value λ = N/c decreases for large N as ∼ exp h − β 2, where β = 1, 2 corresponds respectively to real and complex Wishart matrices, c = N/M ≤ 1 and Φ−(x; c) is a rate (sometimes also called 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 Wishart matrices whose eigenvalues are constrained to be smaller than a fixed barrier. Numerical simulations are in excellent agreement with the analytical predictions.

We consider N × N Gaussian random matrices, whose average density of eigenvalues has the Wigner semicircle form over [-√2],√2]. For such matrices, using a Coulomb gas technique, we compute the large N behavior of the probability P(N,L)(N(L)) that N(L) eigenvalues lie within the box [-L,L]. This probability scales as P(N,L)(N(L) = κ(L)N) ≈ exp(-βN(2)ψ(L)(κ(L))), where β is the Dyson index of the ensemble and ψ(L)(κ(L)) is a β-independent rate function that we compute exactly. We identify three regimes as L is varied: (i) N(-1)≪L < √2 (bulk), (ii) L∼√2 on a scale of O(N(-2/3)) (edge), and (iii) L > sqrt[2] (tail). We find a dramatic nonmonotonic behavior of the number variance V(N)(L) as a function of L: after a logarithmic growth ∝ln(NL) in the bulk (when L∼O(1/N)), V(N)(L) decreases abruptly as L approaches the edge of the semicircle before it decays as a stretched exponential for L > sqrt[2]. This "dropoff" of V(N)(L) at the edge is described by a scaling function V(β) that smoothly interpolates between the bulk (i) and the tail (iii). For β = 2 we compute V(2) explicitly in terms of the Airy kernel. These analytical results, verified by numerical simulations, directly provide for β = 2 the full statistics of particle-number fluctuations at zero temperature of 1D spinless fermions in a harmonic trap.