# LDP for Marchenko Pastur with k/n tending to 0

I am interested in the determinant of $$W = X * X'$$, where $$X \in \mathbb{R}^{k \times n}$$ is a matrix with each row drawn IID from some sub-Gaussian distribution on $$\mathbb{R}^{n}$$. (I am aware of some universality results, so happy to also consider a "standard" Wishart matrix with parameter $$k/n$$). Edit: say e.g. that the diagonal of $$W$$ is all ones, and the off-diagonal are of order $$1/\sqrt n$$.

Question (short): is there an LDP for the empirical spectral distribution of $$W$$ to Marchenko-Pastur when $$k \approx \sqrt{n}$$?

Question (long): There is an argument by Ofer here https://mathoverflow.net/q/372456 that finds the expected determinant of $$W$$, if $$k/n \to c$$ for some constant $$c \in (0,1)$$. The idea is to use an LDP for the convergence of the empirical spectral distribution of Wishart to Marchenko-Pastur.

I am interested in the case of $$k/n \to 0$$, and in particular, the scaling $$k \approx \sqrt{n}$$. Letting $$\lambda := k/n$$, and $$\mu(\lambda)$$ denote the corresponding M-P law, it is easy to check $$\mathbb{E}_{X \sim \mu(k,n)}[\log x] \asymp -\lambda/2 + O(\lambda^2)$$ I would hope that this implies something like $$\mathbb{E}\det(W) = \exp(k \lambda + O(\lambda)) := exp(k^2/n + O(k/n))$$

But if $$k/n \to 0$$, the referenced LDP gives the trivial answer $$\mathbb{E}\det(W) = o(k)$$. Is there a more "quantitative" LDP known that allows me to take $$k$$ and $$n$$ jointly to $$0$$?

• LDP = Large Deviation Principle. Mar 3 at 20:55
• Actually, the argument I gave in that question does not really need the LDP, only concentration, and the latter is known for in your setup (e.g. if entries satisfy log-sobolev, or are bounded). Mar 3 at 22:25
• For a true LDP, unfortunately not much is known for general entries when $k/n\to c$ Mar 3 at 22:28
• For $k/n\to 0$, concentration is even better, so see my first comment. Mar 3 at 22:28
• You are asking many questions at once. In the regime $k/n\to 0$, the $k\times k$ matrix $W$ is very close to $kI_k$ and then it it not hard to check that $E det(W)\sim n^k$ (unless of course you meant somehow to normalize things differently - seems you did because you talk of convergence to MP. Mar 3 at 22:34

For standard Gaussians, and with the matrix $$W/n$$, the proof of the LDP given by Ben Arous-Guionnet adapts to the Wishart setup. However, you will have different scalings and so the non-commutative entropy term (of exponential scaling $$k^2$$) will disappear, and the proof more or less trivializes. If I did not make a stupid computational mistake, the large deviations will have speed $$kn$$ and rate function $$I(\mu)= \frac12 \int (x-\log x) \mu(dx)$$. You see that $$I(\mu)=0$$ iff $$\mu=\delta_1$$.