Universality for random matrices with an edge spectrum singularity

We study invariant random matrix ensembles \begin{equation*} $\mathbb{P}$n(d M)=Z_n^{-1}\exp(-n,tr(V(M))),d M \end{equation*} defined on complex Hermitian matrices $M$ of size $n\times n$, where $V$ is real analytic such that the underlying density of states is one-cut regular. Considering the average \begin{equation*} E_n[φ;λ,α,β]:=$\mathbb{E}$n\bigg(\prod{\ell=1}^n\big(1-φ(λ{\ell}(M))\big)ω_{αβ}(λ_{\ell}(M)-λ)\bigg),\ \ \ \ \ ω_{αβ}(x):=|x|^α\begin{cases}1,&x<0\ β,&x\geq 0\end{cases}, \end{equation*} taken with respect to the above law and where $\phi$ is a suitable test function, we evaluate its large-$n$ asymptotic assuming that $\lambda$ lies within the soft edge boundary layer, and $(\alpha ,\beta )\in \mathbb{R}\times \mathbb{C}$ satisfy $\alpha >-1,\beta \mathrm{\notin }(-\mathrm{\infty },0)$. Our results are obtained by using Riemann-Hilbert problems for orthogonal polynomials and integrable operators and they extend previous results of Forrester and Witte \cite{FW} that were obtained by an application of Okamoto’s $\tau$-function theory. A key role throughout is played by distinguished solutions to the Painlevé-XXXIV equation.