Local universality in biorthogonal Laguerre ensembles

We consider $n$ particles $0\leq x_1<x_2< \cdots < x_n < +\infty$, distributed according to a probability measure of the form $$\frac{1}{Z_n}\prod_{1\leq i <j \leq n}(x_j-x_i)\prod_{1\leq i <j \leq n}(x_j^{\theta}-x_i^{\theta})\prod_{j=1}^nx_j^\alpha e^{-x_j}\ud x_j, ~~ \alpha>-1,~~ \theta>0,$$ where $Z_n$ is the normalization constant. This distribution arises in the context of modeling disordered conductors in the metallic regime, and can also be realized as the distribution for squared singular values of certain triangular random matrices. We give a double contour integral formula for the correlation kernel, which allows us to establish universality for the local statistics of the particles, namely, the bulk universality and the soft edge universality via the sine kernel and the Airy kernel, respectively. In particular, our analysis also leads to new double contour integral representations of scaling limits at the origin (hard edge), which are equivalent to those found in the classical work of Borodin. We conclude this paper by relating the correlation kernels to those appearing in recent studies of products of $M$ Ginibre matrices for the special cases $\theta=M\in\mathbb{N}$.