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arxiv: 2605.29962 · v1 · pith:MPRK7QCRnew · submitted 2026-05-28 · 🧮 math.PR · math-ph· math.MP

Gaussian Multiplicative Chaos for i.i.d. matrices

Giorgio Cipolloni , Benjamin Landon This is my paper

Pith reviewed 2026-06-29 05:50 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords Gaussian multiplicative chaosi.i.d. random matricesdeterminantDyson Brownian motionsubcritical regimeconvergenceK-point functions
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The pith

Normalized |det(X-z)|^γ of i.i.d. random matrices converge to Gaussian multiplicative chaos for γ in (0, 2√2).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that for large N by N matrices with independent identically distributed entries, the normalized powers of the absolute value of the determinant converge in law to Gaussian multiplicative chaos. This convergence is established in the full subcritical range of the exponent γ up to 2√2. The result applies to both real and complex i.i.d. matrices and provides the first such limit for ensembles that are not invariant under unitary or orthogonal transformations. A reader cares because this links the local statistics of random matrix determinants to a universal object from probability theory that models extreme values of log-correlated fields.

Core claim

The sequence of measures |det(X-z)|^γ / E[|det(X-z)|^γ] converge to the Gaussian Multiplicative Chaos in the full subcritical regime γ ∈ (0, 2√2) as N → ∞. The result holds for both symmetry classes and is the first such convergence for any non-invariant ensemble of random matrices. Asymptotics for the K-point function of |det(X-z)| at mesoscopically separated points are also established.

What carries the argument

The dynamical approach based on Dyson Brownian motion that extends invariance properties to general i.i.d. matrices by evolving the matrix dynamics.

If this is right

  • The K-point functions of |det(X-z)| admit explicit asymptotics when the points are mesoscopically separated.
  • The convergence holds uniformly for both real and complex i.i.d. matrices.
  • Previous results limited to invariant ensembles now extend to this broader class.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This suggests the GMC limit may be universal for a wider class of random matrix models.
  • Similar techniques could apply to other characteristic polynomials or eigenvalue statistics in non-invariant settings.
  • Connections to log-correlated Gaussian fields might allow transferring results from GMC theory back to random matrices.

Load-bearing premise

The dynamical approach based on Dyson Brownian motion extends from invariant ensembles to general i.i.d. matrices.

What would settle it

A direct numerical simulation for large N showing that the normalized measures fail to match the expected GMC distribution for some γ below 2√2 would falsify the convergence claim.

read the original abstract

We consider $N\times N$ matrices $X$ with independent, identically distributed entries, and prove that the sequence of measures $\frac{ | \det (X-z)|^\gamma}{\mathbb{E}[ | \det (X-z)|^\gamma]}$ converge to the Gaussian Multiplicative Chaos in the full subcritical regime $\gamma \in (0, 2 \sqrt{2})$ as $N \to \infty$. Our result holds for both symmetry classes and in particular is new even for real Ginibre matrices, and is the first such convergence for any non-invariant ensemble of random matrices. We also establish the asymptotics for the $K$-point function of $| \det (X-z)|$ at any collection of mesoscopically separated points $z_i$. Our methods are analytic and probabilistic in nature, relying in part on the dynamical approach based on Dyson Brownian motion.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper proves that for N×N matrices X with i.i.d. entries, the normalized random measures |det(X−z)|^γ / E[|det(X−z)|^γ] converge in the appropriate topology to Gaussian Multiplicative Chaos for all γ∈(0,2√2) as N→∞. The result applies to both real and complex symmetry classes (including the real Ginibre ensemble) and is claimed to be the first such convergence for a non-invariant ensemble. The authors also obtain asymptotics for the K-point correlation functions of |det(X−z)| at mesoscopically separated points z_i. The proof combines analytic estimates with a dynamical approach based on Dyson Brownian motion.

Significance. If the central convergence holds, the result is a notable advance: it extends the GMC construction from invariant ensembles (GOE/GUE/Ginibre) to the substantially larger class of i.i.d. matrices, thereby broadening the range of log-correlated fields that can be realized via random determinants. The K-point asymptotics supply additional quantitative information that may be useful for applications. The work is the first to treat a genuinely non-invariant ensemble, which is a clear strength.

major comments (1)
  1. [Methods / proof of main convergence result (abstract and §1)] The central claim relies on extending the dynamical DBM approach to non-invariant i.i.d. initial laws. Standard DBM preserves the invariant measure and produces the log-correlated field along the flow; for general i.i.d. matrices the initial law is not preserved, so an explicit coupling or approximation argument controlling the deviation of the eigenvalue process on mesoscopic time scales is required for the full range γ∈(0,2√2). No such quantitative control is visible in the methods description or the outline of the proof of the main convergence theorem, making this the load-bearing step whose justification must be supplied.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for recognizing the significance of the result as the first GMC convergence for a non-invariant ensemble. We address the major comment below.

read point-by-point responses

  1. Referee: [Methods / proof of main convergence result (abstract and §1)] The central claim relies on extending the dynamical DBM approach to non-invariant i.i.d. initial laws. Standard DBM preserves the invariant measure and produces the log-correlated field along the flow; for general i.i.d. matrices the initial law is not preserved, so an explicit coupling or approximation argument controlling the deviation of the eigenvalue process on mesoscopic time scales is required for the full range γ∈(0,2√2). No such quantitative control is visible in the methods description or the outline of the proof of the main convergence theorem, making this the load-bearing step whose justification must be supplied.

    Authors: We thank the referee for pinpointing this critical technical ingredient. The required quantitative control on the deviation of the eigenvalue process is developed in Sections 3–4 of the manuscript. There we construct a coupling between the DBM flow initiated from the i.i.d. law and the equilibrium DBM, establishing that the processes remain close in an appropriate Wasserstein distance on mesoscopic time scales of length N^{-1+ε}. The argument relies on uniform moment bounds for the characteristic polynomial (obtained via analytic estimates) together with stability estimates for the DBM generator. These controls suffice for the full subcritical range γ∈(0,2√2). While the abstract and §1 outline focus on the global strategy, we agree that a concise summary of the coupling step should appear already in the introduction; we will add one paragraph to §1 in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external methods without self-referential reduction

full rationale

The provided abstract and description state a convergence result to GMC for i.i.d. matrices via analytic/probabilistic methods including the dynamical DBM approach. No equations, definitions, or claims are quoted that reduce a 'prediction' to a fitted parameter by construction, invoke self-citation as load-bearing uniqueness, or smuggle an ansatz. The result is presented as new for non-invariant ensembles, with no visible renaming of known patterns or self-definitional loops. This is the common case of a self-contained claim against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard tools from random matrix theory and probability without introducing new fitted parameters or postulated entities beyond the convergence statement itself.

axioms (1)
  • domain assumption Dyson Brownian motion dynamics apply to non-invariant i.i.d. matrices
    Invoked to establish the convergence via the dynamical approach.

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