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In this paper, we study the singularly perturbed Laguerre unitary ensemble $$\frac{1}{Z_n} (\det M)^\alpha e^{- \textrm{tr}\, V_t(M)}dM, \qquad \alpha >0,$$ with $V_t(x) = x + t/x$, $x\in (0,+\infty)$ and $t>0$. Due to the effect of$t/x$ for varying $t$, the eigenvalue correlation kernel has a new limitinstead of the usual Bessel kernel at the hard edge 0. This limiting kernelinvolves $\psi$-functions associated with a special solution to a newthird-order nonlinear differential equation, which is then shown equivalent toa particular Painlev\'e III equation. The transition of this limiting kernel tothe Bessel and Airy kernels is also studied when the parameter $t$ changes in afinite interval $(0, d]$. Our approach is based on Deift-Zhou nonlinearsteepest descent method for Riemann-Hilbert problems.

Critical edge behavior and the Bessel to Airy transition in the singularly perturbed Laguerre unitary ensemble