What does one-cut random matrix mean?

Question

I am quite new to random matrix theory and recently I encountered the so-called "one-cut random matrix model" and even "two-cut" in physics. So what exactly does it mean?

Answer 1

The name 'one-cut' or 'two-cut' has to do with the support of the limiting spectral density as $N\to\infty$, where $N$ is the size of your matrix. Take for example a $N\times N$ real symmetric matrix whose entries in the upper triangle are independent Gaussian variables with mean zero and variance $1$ [there is a small subtlety on the variance of diagonal elements, but it is unimportant now]. Collect the $N$ real eigenvalues $\{\lambda_1^{(1)},\ldots,\lambda_N^{(1)}\}$ for this first sample (hence the superscript $^{(1)}$). Now, repeat the procedure $M$ times, and collect all the eigenvalues (rescaled by $\sqrt{N}$) of the $M$ samples in a single string of size $N\times M$ $$ \{\lambda_1^{(1)}/\sqrt{N},\ldots,\lambda_N^{(1)}/\sqrt{N},\ldots,\lambda_1^{(M)}/\sqrt{N},\ldots,\lambda_N^{(M)}/\sqrt{N}\}\ . $$ Now produce a normalized histogram of this string. For "sufficiently large" $N$ and $M$, your normalized histogram will converge to a "semicircular" (or rather semielliptical) shape (the celebrated Wigner's semicircular law), whose analytical expression is $$ \rho(\lambda) = \frac{1}{\pi}\sqrt{2-\lambda^2}\ . $$ Note that this average spectral density is supported on $\sigma=[-\sqrt{2},\sqrt{2}]$, i.e. a single interval of the real line. As a consequence, the Cauchy–Stieltjes transform $G(z)$ of $\rho(\lambda)$ $$ G(z)=\int_\sigma d\lambda\frac{\rho(\lambda)}{z-\lambda} $$

has one connected cut on the real line of the complex plane, hence the name one-cut. Multiple-cut ensembles are just characterized by the property that in the limit $N\to\infty$ the eigenvalues aggregate in disconnected "blobs", each of which is supported on an interval of the real line - but the blobs themselves are separated by "empty" intervals (almost surely, in between these blobs, no eigenvalue will be found).