A mathematical preprint dated 28 August 2026 reports a family of limiting behaviors for the characteristic polynomials of self-normalized random matrices. As the tail index changes, the authors describe the family as moving from Poisson multiplicative chaos at alpha=0 toward Gaussian multiplicative chaos as alpha approaches 2. The work is identified in its front matter as an arXiv preprint, version v1.
The model begins with a row-wise reset
The model starts with a data matrix whose entries are independent and identically distributed, then normalizes each row by its Euclidean length. The resulting rows are independent and each has unit Euclidean norm. Before normalization, the absolute values of the entries are assumed to have a regularly varying tail with index alpha and a slowly varying factor. In this analysis, alpha is the parameter that tracks the tail regime.
For tail indices strictly between 0 and 2, the characteristic-polynomial function converges in distribution on the open unit disk as the matrix dimension grows. In ordinary terms, the random function approaches a limiting analytic function, called F_alpha, across the disk rather than only at one chosen point. The convergence is local-uniform, the paper's way of stating that this function-level behavior is controlled throughout the unit-disk domain.
The limit belongs to the class the authors call alpha-heavy multiplicative chaos. It is described as the exponential of a random series, so the result identifies a random analytic function rather than a fixed number.
Traces carry the same signature
Traces of powers of the matrix show the same tail-index signature. For any fixed finite collection of positive integer powers, the corresponding trace vector converges jointly in distribution to the matching alpha-Poisson-family variables. Joint convergence means the several trace quantities are treated together in the same limit, rather than one at a time. The proof studies centered trace variables and uses the method of moments.
At the level of functions, the convergence argument combines tightness with convergence of coefficients. Those are the two ingredients used to move from coefficient-level information to a limit for the random analytic functions.
The spectral-radius result has a narrow edge
The paper also gives a one-sided result for the spectral radius, meaning the largest absolute value of a matrix's eigenvalues. For every fixed positive epsilon, the probability that the spectral radius is at least 1 plus epsilon tends to zero as the dimension grows. Put simply, eigenvalues beyond one by any fixed margin become unlikely in the large-dimension limit.
That result is an upper bound, not a proof that the spectral radius converges to one. The analysis supplies no matching lower bound, leaving convergence to one as a conjecture rather than an established conclusion. Because the result is asymptotic, it should not be read as a finite-dimension guarantee that the radius is at most one.
The endpoints change the character of the limit
At the lower endpoint, alpha=0, where the paper considers slowly varying tails, the characteristic polynomial converges in distribution to the Poisson-endpoint function F0. At the Gaussian boundary, alpha=2, the result is under the paper's domain-of-attraction-of-the-normal, or DAN, assumption. There, the limiting characteristic polynomial is expressed through independent standard Gaussian variables.
Together, the interior and boundary results underpin the authors' description of a Poisson-to-Gaussian interpolation. The interior trace limits remain in the alpha-Poisson family, while the Gaussian boundary has a Gaussian characteristic-polynomial form. The supplied results do not say that a fixed interior tail index has a Gaussian trace limit.
A simpler matrix offers a separate comparison
One additional comparison focuses on a one-hot matrix. Under condition (2.7), the row-normalized matrix approaches that one-hot matrix in Frobenius norm in probability. The Frobenius norm measures the combined size of the entry-by-entry differences, so the theorem treats the two matrices as becoming close in probability as dimension grows.
For one-hot matrices, the associated spectral measure converges in probability to a point mass at zero. A spectral measure records the distribution of the matrix's eigenvalues, and a point mass at zero means the limiting measure is concentrated there. This result is specifically about the one-hot matrices; it is not a proof that every self-normalized matrix in the heavy-tailed range has the same spectral-measure limit.
A result tied to its assumptions
The theorem-level conclusions are conditional on the stated model: independent, identically distributed, symmetric entries, with the required regularly varying tail or normal-domain assumption. The supplied work is asymptotic, so its statements concern what happens as the matrix dimension tends to infinity.
The work is identified as an arXiv preprint, version v1, dated 28 August 2026. The supplied text reports partial Swedish Research Council support for Johannes Heiny and Xuechun Hu through grant VR-2023-03577.
Paper data and sources
Original title: Characteristic polynomial of self-normalized random matrices
Authors: Quentin François, Johannes Heiny, Xuechun Hu
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text