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[\phi;\lambda,\alpha,\beta]:=\mathbb{ E}_n\bigg(\prod_{ \ell=1}^n\big(1-\phi(\lambda_{ \ell}(M))\big)\omega_{ \alpha\beta}(\lambda_{ \ell}(M)-\lambda)\bigg),\ \ \ \ \ \omega_{ \alpha\beta}(x):=|x|^{ \alpha}\begin{ cases}1,&x<0\\ \beta,&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 $\alpha>-1,\beta\notin(-\infty,0)$. 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 Painlev\'e-XXXIV equation.
Categories: math-ph math.CA math.MP nlin.SI