Critical characteristic-polynomial chaos for complex iid matrices

@misc{cai2026critical,
  title = {{Critical characteristic-polynomial chaos for complex iid matrices}},
  author = {Yin Cai and Xiang Fang and Feng Guo},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01794},
  eprint = {2611.01794},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01794},
}

TY  - UNPB
TI  - Critical characteristic-polynomial chaos for complex iid matrices
AU  - Yin Cai
AU  - Xiang Fang
AU  - Feng Guo
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01794
UR  - https://arxiv.org/abs/2611.01794
AB  - We prove convergence to critical Gaussian multiplicative chaos for characteristic-polynomial measures of complex iid matrices. Let A_N=N^−1/2(ξ_ij), with a fixed law satisfying 𝔼ξ=𝔼ξ^2=0, 𝔼|ξ|^2=1, and finite moments of every order; no density is assumed. On each fixed disk {|z|≤ r}, 0<r<1, expectation-normalized determinant measures at power 2√2, multiplied by √(1/2log N), converge to critical chaos with a fourth-cumulant covariance correction. The normalized height-deficit marked measure converges to the product of this spatial limit and the standard Rayleigh law. Convergence holds jointly with centered averages of the √2-scaled original logarithmic field against finitely many smooth compactly supported bulk tests, and with the two-sided near-critical curve at powers q=2√2−2λ/√(log N). On fixed compact parameter intervals, the curve converges in law in the uniform weak topology; all limiting coordinates use the same Gaussian field. The limiting curve is an explicit deterministic profile times the same critical measure.
ER  -