Critical characteristic-polynomial chaos for complex iid matrices

Abstract

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 $\mathbb Eξ=\mathbb Eξ^2=0$, $\mathbb E|ξ|^2=1$, and finite moments of every order; no density is assumed. On each fixed disk $\{|z|\le r\}$, $0<r<1$, expectation-normalized determinant measures at power $2\sqrt2$, multiplied by $\sqrt{\tfrac12\log N}$, 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 $\sqrt2$-scaled original logarithmic field against finitely many smooth compactly supported bulk tests, and with the two-sided near-critical curve at powers $q=2\sqrt2-2λ/\sqrt{\log N}$. 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.