The classical $\beta$-ensembles with $\beta$ proportional to $1/N$: from loop equations to Dyson's disordered chain

Peter J. Forrester, Guido Mazzuca

In the classical $\beta$-ensembles of random matrix theory, setting $\beta =2\alpha /N$ and taking the $N\to \infty$ limit gives a statistical state depending on $\alpha$. Using the loop equations…