Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

AbstractIn this paper, we study the singularly perturbed Laguerre unitary ensemble $$\frac{1}{Z_n} ({\rm det}\,\, M)^\alpha e^{- {\rm tr}\, V_t(M)}dM, \qquad \alpha > 0,$$1Zn(detM)αe-trVt(M)dM,α>0,with $${V_t(x) = x + t/x,\,\, x \in (0,+\infty)}$$Vt(x)=x+t/x,x∈(0,+∞) and t >  0. Due to the effect of t/x for varying t, the eigenvalue correlation kernel has a new limit instead of the usual Bessel kernel at the hard edge 0. This limiting kernel involves $${\psi}$$ψ-functions associated with a special solution to a new third-order nonlinear differential equation, which is then shown to be equivalent to a particular Painlevé III equation. The transition of this limiting kernel to the Bessel and Airy kernels is also studied when the parameter t changes in a finite interval (0, d]. Our approach is based on Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems.