Properties of the chGUE at the Hard Edge: Spacing Distributions and Universality with External Field

The chiral Gaussian unitary ensemble also called Laguerre or Wishart unitary ensemble is probably one of the most studied random matrix ensembles. We will investigate its further properties at the hard edge of the spectrum at finite- and large-N. Due to applications to the Dirac operator in Quantum Chromodynamics, we will add a finite number $N_f$ of characteristic polynomials to the Gaussian distribution of matrix elements. In the first part we will focus on the spacing distribution between the smallest singular values, where we give exact determinantal formulae. In contrast to the k-point correlation functions, the spacing is almost immediately as close to the bulk spacing as the Wigner surmise. In the second part we will show that the k-point functions are universal at the hard edge under the addition of an external, deterministic field $A$ with full rank, as long as a hard edge is present. Using recent progress in polynomial ensembles we can show that previous results from supersymmetry with and orthogonal polynomials without external field agree and can be extended to a fixed number of zero modes. In particular we show that determinantal formulae of different sizes for the k-point functions are equivalent.

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