RT Journal Article
T1 Characteristic polynomials of complex random matrices and Painlevé transcendents
A1 Deaño Cabrera, Alfredo
A1 Simm, Nick
AB We study expectations of powers and correlation functions for characteristic polynomials of N×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→∞⁠. 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→∞⁠. 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.
PB Oxford University Press
SN 1073-7928
YR 2021
FD 2021-12
LK https://hdl.handle.net/10016/33852
UL https://hdl.handle.net/10016/33852
LA eng
DS e-Archivo
RD 1 sept. 2024