How are the saddle-point equations derived in the single random matrix model?

Question

In this question, I'm referring to a specific step in https://arxiv.org/abs/hep-th/9306153. I want to reproduce equation (2.4) on page 15. I think I lack the experience required for dealing with saddle-point approximations to multi-variable integrals such as the one appearing in (2.3). I'm approaching this having an understanding of the saddle-point approximation as I've seen it previously in physics, roughly what's described in the first answer to: How is the Saddle point approximation used in physics?.

Here's how I've tried to think about it so far,

\begin{align} \int \prod_{i} \mathrm{d} \lambda_{i} \Delta^{2}(\lambda) \mathrm{e}^{-(N / g) \sum_{i} V\left(\lambda_{i}\right)} = e^{(N / g)}\int \mathrm{d} \lambda_{0}\mathrm{d} \lambda_{1}...\mathrm{d} \lambda_{N} \Delta^{2}(\lambda) \mathrm{e}^{-V\left(\lambda_{0}\right)-V\left(\lambda_{1}\right)-\ \dotsm \ -V\left(\lambda_{N}\right)}. \end{align}

I should note here that I believe there is a typo in the review, and that the argument to the Vandermonde determinant should be $\lambda$ as it is written here, and not $\Lambda$ as it appears in the review.

Now I could perform the saddle-point approximation for the integral over each eigenvalue, $\lambda_i$. A technical issue will be how I deal with the Vandermonde determinant, and I wonder if the sentence,

The Vandermonde determinant leads to a repulsive force between eigenvalues which otherwise would accumulate at the minimum of the potential $V$.

is somehow related to that issue. I want to try and think about the derivation for a simple case, like $N=3$, but the whole thing is supposed to happen in the large $N$ limit, so that seems like an obvious mistake.

Further, I don't even understand conceptually why there should be a set of "saddle-point equations" that are derived from "varying a single eigenvalue". Perhaps someone can direct me to a simpler example of the techniques used here, and I could work on understanding that first.

Answer 1

The partition function for the random matrix model in 0+0D is of the form $$ Z~=~\left[\prod_{i=1}^N\int_{\mathbb{R}}\!\mathrm{d}\lambda_i\right]e^{-S(\lambda)},\tag{2.1}$$ with Euclidean action $$S(\lambda)~=~\frac{N}{g} V(\lambda)+V_{\rm Vandermonde}(\lambda), \qquad V_{\rm Vandermonde}(\lambda)~=~-2\sum_{1\leq i<j\leq N} \ln|\lambda_i-\lambda_j|.$$ Note that the second term, the Vandermonde potential becomes smallest if the eigenvalues are spread out, i.e. it is repulsive. The EL equations are the sought-for eq. (2.4): $$0~\approx~\frac{\partial S(\lambda)}{\partial \lambda_k}~=~\frac{N}{g}\frac{\partial V(\lambda)}{\partial \lambda_k} -\sum_{j\neq k}\frac{1}{\lambda_k-\lambda_j} .\tag{2.4}$$