Multi-Cut Matrix Models

Question

I have a question pertaining specifically to a one-matrix model with a multi-cut solution. The standard procedure is to take a polynomial superpotential $W(x)$. In the classical limit (analogous to $\hbar =0$), all of the eigenvalues are sitting precisely at the extrema of $W$. As you turn some sort of coupling on, and pass to the large-$N$ 't Hooft limit (the matrices are $N \times N$), the eigenvalues "repel" each other and spread out to form "branch cuts."

I had been assuming that this spreading out from the extrema of $W$ happens symmetrically, but perhaps they won't spread symmetrically if the eigenvalues from the other cuts can repel them? To make my question more concrete, imagine just two extrema of $W$ at $\pm1$ on the real axis. Let's assume the cuts will spread out also on the real axis. Will each cut emerge symmetrically from $\pm1$? Or will they actually spread more on the "outside" since the eigenvalues might repel each other more on the "inside"?

I haven't seen this discussed in the matrix model literature so I was hoping perhaps someone had some insight!

This post imported from StackExchange Physics at 2016-06-26 09:55 (UTC), posted by SE-user spietro