Hard edge universality of Muttalib-Borodin ensembles with real parameter $\theta$

We analyse the hard edge limit of the Muttalib-Borodin ensembles with general potential, and show that the limiting correlation kernel found in the ensemble with linear potential is universal. We also prove the Plancherel-Rotach type asymptotics of the biorthogonal polynomials associated to the Muttalib-Borodin ensembles around zero, where the limits are given by Wright's generalized Bessel functions. To accomplish these results, we implement the Deift-Zhou steepest-descent method on the vector Riemann-Hilbert problems for the biorthogonal polynomials, and develop a new method to construct the hard edge local parametrix at zero. The results in this paper are valid for all real parameter $\theta > 0$ in the Muttalib-Borodin ensembles, and this paper generalizes [Wang-Zhang21] that considers only the integer $\theta$ case.