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.


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): $$0~\approx~\frac{\partial S(\lambda)}{\partial \lambda_k}~=~\frac{N}{g}\frac{\partial V(\lambda)}{\partial \lambda_k} -\sum_{j\neq k}\frac{1}{\lambda_k-\lambda_j} .\tag{2.4}$$

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.