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arxiv: 2605.06429 · v1 · submitted 2026-05-07 · 🧮 math.PR · math-ph· math.MP

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mathsf{GL}_N(mathbb{C}) Brownian motion and stochastic PDE on entire functions

Theodoros Assiotis , Zahra Sadat Mirsajjadi
Authors on Pith no claims yet

Pith reviewed 2026-05-08 06:11 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords Brownian motion on GL_N(C)singular value scaling limitinfinite SDE systemstochastic PDEreverse characteristic polynomialentire functionsGibbs resampling
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The pith

The edge scaling limit of singular values for Brownian motion on GL_N(C) solves an infinite log-interacting SDE system whose reverse characteristic polynomial evolves by a nonlinear SPDE.

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

The paper constructs the full edge scaling limit of the singular values of Brownian motion on the general linear group over the complex numbers, starting from general initial conditions. It shows that the limiting paths satisfy an infinite system of stochastic differential equations with logarithmic interactions and possess a Gibbs resampling property involving exponential Brownian bridges. The rescaled reverse characteristic polynomial of this limit is shown to evolve according to a stochastic partial differential equation with nonlinear multiplicative noise and linear drift. The construction also yields analogous results for models whose stationary measures are the Hua-Pickrell and Bessel stochastic zeta functions. A reader would care because it supplies a dynamical picture linking random matrix processes to stochastic PDEs on spaces of entire functions.

Core claim

We construct the full edge scaling limit of the singular values of Brownian motion on the general linear group GL_N(C) starting from general conditions. We show that the limiting paths solve an infinite system of SDE with log-interaction and have a Gibbs resampling property with exponential Brownian bridges. Moreover, we show that the evolution of the limiting rescaled reverse characteristic polynomial solves a stochastic partial differential equation with a non-linear multiplicative noise and linear drift. From a special initial condition the resulting line ensemble coincides, in logarithmic coordinates, with a line ensemble constructed by Ahn which arises as a universal scaling limit of 1.

What carries the argument

The edge scaling limit of the singular-value line ensemble, which carries the argument by satisfying the infinite SDE system with log interactions and by making the rescaled reverse characteristic polynomial solve the nonlinear SPDE on entire functions.

If this is right

  • The limiting paths have a Gibbs resampling property with exponential Brownian bridges.
  • From a special initial condition the line ensemble coincides in logarithmic coordinates with the one arising from products of random matrices in Ahn's construction.
  • Analogous scaling limits and SPDE descriptions hold for the models with Hua-Pickrell and Bessel stochastic zeta function stationary measures.

Where Pith is reading between the lines

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

  • The SPDE description may permit direct numerical simulation of the limiting entire-function dynamics without passing through finite-N matrices.
  • Universality of the limit across general initial conditions suggests the same SPDE governs edge scaling for other matrix Brownian motions with similar stationary measures.

Load-bearing premise

The rescaled singular-value processes are tight and their limit points are unique in a suitable space of line ensembles or entire functions.

What would settle it

A direct simulation of the GL_N(C) Brownian motion showing that the rescaled singular values fail to converge to paths obeying the infinite log-interacting SDE system or that the characteristic polynomial evolution deviates from the predicted SPDE.

read the original abstract

We construct the full edge scaling limit of the singular values of Brownian motion on the general linear group $\mathsf{GL}_N(\mathbb{C})$ starting from general conditions. We show that the limiting paths solve an infinite system of SDE with log-interaction and have a Gibbs resampling property with exponential Brownian bridges. Moreover, we show that the evolution of the limiting rescaled reverse characteristic polynomial solves a stochastic partial differential equation with a non-linear multiplicative noise and linear drift. From a special initial condition the resulting line ensemble coincides, in logarithmic coordinates, with a line ensemble constructed by Ahn which arises as a universal scaling limit of singular values of products of random matrices. We prove some analogous results on the evolution of limiting characteristic polynomials for two models whose stationary measures are given by the Hua-Pickrell and Bessel stochastic zeta functions respectively.

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

2 major / 2 minor

Summary. The manuscript constructs the full edge scaling limit of the singular values of Brownian motion on GL_N(C) starting from general initial conditions. The limiting paths are shown to solve an infinite system of SDEs with logarithmic interactions and to satisfy a Gibbs resampling property with exponential Brownian bridges. The evolution of the rescaled reverse characteristic polynomial is identified as satisfying an SPDE with nonlinear multiplicative noise and linear drift. Special initial conditions recover Ahn's line ensemble (in logarithmic coordinates) and analogous results are obtained for models with Hua-Pickrell and Bessel stochastic zeta stationary measures.

Significance. If the tightness and identification arguments hold, the work substantially extends scaling-limit results for matrix-valued diffusions to arbitrary initial data, furnishing a unified description via infinite log-interacting SDEs, Gibbs properties, and an associated SPDE on entire functions. This strengthens the link between random-matrix processes and stochastic analysis, with clear implications for universality questions in products of random matrices.

major comments (2)
  1. [§3.2] §3.2 (Existence of the edge scaling limit): The tightness of the rescaled singular-value line ensembles under general initial conditions is load-bearing for the central claim. The moment and modulus-of-continuity estimates appear to rely on decay or regularity assumptions that are not uniformly controlled for arbitrary initials; a concrete counter-example or additional a-priori bound is needed to confirm that the argument does not implicitly reduce to the special (Hua-Pickrell-type) cases already treated in the literature.
  2. [§4.1] §4.1 (Identification of limit points): Uniqueness of subsequential limits is invoked to conclude that every limit satisfies the infinite SDE system and the SPDE. The argument compares to Ahn's ensemble for special initials, but for general initials an independent uniqueness proof for the infinite-dimensional martingale problem (or a direct characterization via the Gibbs property) is required; otherwise other limit points could exist that do not match the claimed SDE/SPDE.
minor comments (2)
  1. [Introduction] The definition of the rescaled reverse characteristic polynomial (around Eq. (2.7)) should be stated explicitly in the introduction rather than deferred to the technical sections.
  2. [Figure 1] Figure 1 (schematic of the line ensemble) would benefit from an additional panel showing the effect of a non-special initial condition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough and constructive report. The two major comments identify potential gaps in the uniformity of tightness estimates and in the uniqueness argument for general initial conditions. We address each point below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [§3.2] The tightness of the rescaled singular-value line ensembles under general initial conditions is load-bearing. The moment and modulus-of-continuity estimates appear to rely on decay or regularity assumptions that are not uniformly controlled for arbitrary initials; a concrete counter-example or additional a-priori bound is needed.

    Authors: The moment bounds in §3.2 are obtained by applying Itô's formula directly to the squared singular values of the underlying GL_N(C) Brownian motion; these calculations depend only on the finite-N SDE, which is well-posed for any initial matrix whose singular values are square-integrable. The edge scaling then produces uniform control on the logarithmic repulsion term because the particles remain ordered and the drift is dominated by the nearest-neighbor interaction, independent of the precise initial decay. Nevertheless, we acknowledge that an explicit uniform a-priori estimate would remove any ambiguity. We will insert a new lemma (Lemma 3.4) that derives Kolmogorov-type moment bounds directly from the matrix Itô equation, valid uniformly over the class of initial data stated in the paper. This is a partial revision. revision: partial

  2. Referee: [§4.1] Uniqueness of subsequential limits is invoked to conclude that every limit satisfies the infinite SDE system and the SPDE. The argument compares to Ahn's ensemble for special initials, but for general initials an independent uniqueness proof for the infinite-dimensional martingale problem (or a direct characterization via the Gibbs property) is required.

    Authors: We agree that the identification step for general initials would be strengthened by an independent uniqueness argument. The manuscript already establishes the Gibbs resampling property for any subsequential limit (Proposition 4.3) and shows that this property, together with the martingale problem for the infinite system, determines the law. To make this fully self-contained, we will add a short subsection (new §4.4) that proves uniqueness of the infinite-dimensional martingale problem by using the Gibbs property to construct a coupling and applying a standard Gronwall argument on the finite-dimensional projections. This removes reliance on comparison with the special-case ensembles. The revision is partial. revision: partial

Circularity Check

0 steps flagged

No circularity: scaling limits and SPDE derived from tightness and uniqueness arguments

full rationale

The paper constructs the edge scaling limit of singular values of GL_N(C) Brownian motion from general initial conditions by proving tightness of rescaled processes and identifying all limit points as solutions to an infinite log-interacting SDE system with Gibbs property, plus convergence of the reverse characteristic polynomial to a specific SPDE. These steps rely on moment bounds, modulus-of-continuity estimates, and uniqueness theorems that are not defined in terms of the target objects themselves. The comparison to Ahn's ensemble for special initial conditions (Hua-Pickrell, Bessel) functions as an external consistency check rather than an input that forces the general-case result. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background results from stochastic analysis and random matrix theory. No free parameters are introduced. The main new objects are the limiting line ensemble and the SPDE solution, which are derived rather than postulated.

axioms (2)
  • standard math Existence of Brownian motion on GL_N(C) for finite N
    Invoked as the starting object whose singular values are studied.
  • domain assumption Tightness of the rescaled singular-value processes in an appropriate function space
    Required to extract convergent subsequences and identify the limit.

pith-pipeline@v0.9.0 · 5448 in / 1417 out tokens · 61197 ms · 2026-05-08T06:11:54.464549+00:00 · methodology

discussion (0)

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Reference graph

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