Characteristic Polynomials of Complex Random Matrices and Painlevé Transcendents

Abstract

We study expectations of powers and correlation functions for characteristic polynomials of $N \times N$ non-Hermitian random matrices. For the $1$-point and $2$-point correlation function, we obtain several characterizations in terms of Painlevé transcendents, both at finite $N$ and asymptotically as $N \to \infty $. In the asymptotic analysis, two regimes of interest are distinguished: boundary asymptotics where parameters of the correlation function can touch the boundary of the limiting eigenvalue support and bulk asymptotics where they are strictly inside the support. For the complex Ginibre ensemble this involves Painlevé IV at the boundary as $N \to \infty $. Our approach, together with the results in [ 49], suggests that this should arise in a much broader class of planar models. For the bulk asymptotics, one of our results can be interpreted as the merging of two “planar Fisher–Hartwig singularities” where Painlevé V arises in the asymptotics. We also discuss the correspondence of our results with a normal matrix model with $d$-fold rotational symmetries known as the lemniscate ensemble, recently studied in [ 15, 18]. Our approach is flexible enough to apply to non-Gaussian models such as the truncated unitary ensemble or induced Ginibre ensemble; we show that in the former case Painlevé VI arises at finite $N$. Scaling near the boundary leads to Painlevé V, in contrast to the Ginibre ensemble.

Asymptotics of determinants with a rotation-invariant weight and discontinuities along circles

“…This fact is well-known and has already been used in different contexts, see e.g. [35,21,14,18]. For convenience, we also give a proof of this result here.…”

“…they considered the case of a Gaussian weight perturbed with a planar root-type singularity located in the bulk. Deaño and Simm in [14] then investigated the "edge regime" when n → +∞ and simultaneously |z 0 | → 1 at a critical speed. The case of two merging planar root-type singularities in the bulk was also studied in [14], among other things.…”

“…Deaño and Simm in [14] then investigated the "edge regime" when n → +∞ and simultaneously |z 0 | → 1 at a critical speed. The case of two merging planar root-type singularities in the bulk was also studied in [14], among other things. We also mention that the related topic of planar orthogonal polynomials associated with a Gaussian weight having root-type singularities has been studied in [4,29,30,31], see also [1].…”

Characteristic polynomials of random truncations: moments, duality and asymptotics

“…Such dualities have been observed in several settings in random matrix theory, most frequently in the case of Hermitian or Circular ensembles [11, 12, 18-20, 24, 28]. For non-Hermitian ensembles, related dualities are known in the complex case [3,17,33], especially for the complex Ginibre ensemble [40,49].…”

“…Remark 1.5. In the self-dual case β ′ = β = 2, the identity (1.5) and the asymptotics of Proposition 1.4 were obtained in [17] using the method of orthogonal polynomials. Here we recover this result with a different approach and generalise it to β ∈ {1, 2, 4}.…”