arxiv: 2604.19738 · v2 · submitted 2026-04-21 · 🧮 math.PR · cs.LG· stat.ML

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Phase Transitions in the Fluctuations of Functionals of Random Neural Networks

Simmaco Di Lillo , Leonardo Maini , Domenico Marinucci

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classification 🧮 math.PR cs.LGstat.ML MSC 60F0560G6060H07

keywords random neural networkslimit theoremsWiener chaoscovariance functionsGaussian fieldsphase transitionsHermite expansionsStein-Malliavin

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The pith

The asymptotic behavior of functionals of infinitely-wide random neural networks on the sphere splits into three regimes based on covariance fixed points as depth increases.

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

The paper proves limit theorems for functionals applied to the outputs of random neural networks that are infinitely wide and produce Gaussian fields on the d-dimensional sphere. It shows that as the depth grows, the distributions of these functionals fall into one of three categories depending on the fixed points of the covariance function: matching the functional of a limiting Gaussian field, becoming Gaussian, or landing in a Wiener chaos of order Q. Sympathetic readers would care because this provides a classification of how depth affects the statistics of such networks, highlighting the role of the covariance operator's iterative fixed points in determining convergence types.

Core claim

We establish central and non-central limit theorems for sequences of functionals of the Gaussian output of an infinitely-wide random neural network on the d-dimensional sphere. We show that the asymptotic behaviour of these functionals as the depth of the network increases depends crucially on the fixed points of the covariance function, resulting in three distinct limiting regimes: convergence to the same functional of a limiting Gaussian field, convergence to a Gaussian distribution, convergence to a distribution in the Qth Wiener chaos. The proofs exploit classical tools like Hermite expansions, Diagram Formula, Stein-Malliavin techniques, and the fixed-point structure of the iterative op

What carries the argument

The fixed-point structure and stability of the iterative operator associated with the covariance function, which governs the three limiting regimes.

Load-bearing premise

The fixed-point structure and stability of the iterative operator associated with the covariance function fully determines the three distinct limiting regimes for the functionals.

What would settle it

A covariance function whose fixed points produce a limiting distribution of a functional that matches none of the three regimes would falsify the classification.

read the original abstract

We establish central and non-central limit theorems for sequences of functionals of the Gaussian output of an infinitely-wide random neural network on the d-dimensional sphere . We show that the asymptotic behaviour of these functionals as the depth of the network increases depends crucially on the fixed points of the covariance function, resulting in three distinct limiting regimes: convergence to the same functional of a limiting Gaussian field, convergence to a Gaussian distribution, convergence to a distribution in the Qth Wiener chaos. Our proofs exploit tools that are now classical (Hermite expansions, Diagram Formula, Stein-Malliavin techniques), but also ideas which have never been used in similar contexts: in particular, the asymptotic behaviour is determined by the fixed-point structure of the iterative operator associated with the covariance, whose nature and stability governs the different limiting regimes.

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.

Desk Editor's Note private letter to a colleague

The paper ties fluctuation limits for NN functionals to covariance fixed-point stability in a clean way, but the mapping may need uniformity conditions over functionals.

read the letter

The core contribution is a classification of asymptotic regimes for functionals of the Gaussian field produced by an infinite-width random neural network on the sphere. As depth grows, the behavior splits into three cases—convergence to the same functional of a limiting field, convergence in distribution to a Gaussian, or convergence to a random variable in the Qth Wiener chaos—depending on whether the relevant fixed point of the covariance iteration is attractive, marginal, or repulsive. The proofs combine standard Hermite expansions and Stein-Malliavin bounds with an analysis of the iterative operator on the covariance kernel, which is the part that feels new in this setting. That operator analysis organizes the decay rates of the covariances and thereby selects the leading chaos order, which is a useful organizing principle for this model. The derivations look technically competent and rest on classical tools applied to the network recursion, with no obvious circularity or heavy self-citation. The sphere setting and infinite-width Gaussian output are handled explicitly, which keeps the setup clean. One soft spot is the scope over functionals. The leading chaos degree Q is determined by the lowest non-vanishing projection of the functional onto the fluctuations of the field. If a given functional annihilates the first few chaos components that are sensitive to the covariance decay, the effective regime could shift or the limit could become degenerate even when the fixed-point stability predicts a particular Q. The abstract presents the three regimes as governed by the fixed-point structure without spelling out uniformity conditions, so the proofs should make clear whether the results hold for generic functionals or require the lowest projection to align with the fluctuation rate. Minor gaps like that are common in this style of work and do not sink the main claims. This is for probabilists and theoretical machine-learning researchers who care about limit theorems for random fields and deep networks. A reader working on Gaussian processes or Malliavin calculus on spheres would find the fixed-point perspective useful. The paper has enough new structure and concrete limit theorems to deserve referee time, even if the generality statement needs tightening.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes central and non-central limit theorems for sequences of functionals of the Gaussian output of an infinitely-wide random neural network on the d-dimensional sphere. It demonstrates that the asymptotic behavior of these functionals as the network depth increases is governed by the fixed points of the covariance function, leading to three distinct limiting regimes: convergence to the functional of a limiting Gaussian field, convergence to a Gaussian distribution, or convergence to a distribution in the Qth Wiener chaos. The proofs combine classical tools such as Hermite expansions, the Diagram Formula, and Stein-Malliavin techniques with a novel fixed-point analysis of the iterative operator associated with the covariance function.

Significance. If the central claims hold, the work provides a significant advancement in understanding phase transitions in the fluctuations of neural network functionals through the lens of covariance fixed-point stability. The integration of fixed-point analysis with probabilistic limit theorems offers a new perspective that could influence studies in random neural networks and Gaussian processes. The manuscript credits the use of established tools alongside original ideas in the fixed-point structure.

major comments (2)

[Abstract] The abstract states that the asymptotic behaviour 'depends crucially on the fixed points of the covariance function, resulting in three distinct limiting regimes'. However, the stability of the fixed point controls the decay rate of Cov_l(x,y) - K^*(x,y), but the leading chaos degree Q for a general functional is determined by its lowest non-vanishing Hermite projection. For functionals that annihilate lower-order components (e.g., even functionals when fluctuations are odd), the effective regime may not align with the predicted one, requiring additional uniformity conditions on the functional class.
[Main results (presumed §3 or similar)] The derivation relies on the fixed-point structure fully determining the regimes via Hermite expansion, but without explicit verification that the mapping is one-to-one for arbitrary functionals, the central claim risks overgeneralization. A concrete counterexample or additional assumption on the functional would strengthen this.

minor comments (2)

[Abstract] The phrase 'ideas which have never been used in similar contexts' could be softened to 'novel applications' to avoid overstatement.
[Notation] Clarify the definition of the iterative operator associated with the covariance early in the paper for reader accessibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and have revised the manuscript to enhance clarity on the roles of fixed-point stability and Hermite rank.

read point-by-point responses

Referee: [Abstract] The abstract states that the asymptotic behaviour 'depends crucially on the fixed points of the covariance function, resulting in three distinct limiting regimes'. However, the stability of the fixed point controls the decay rate of Cov_l(x,y) - K^*(x,y), but the leading chaos degree Q for a general functional is determined by its lowest non-vanishing Hermite projection. For functionals that annihilate lower-order components (e.g., even functionals when fluctuations are odd), the effective regime may not align with the predicted one, requiring additional uniformity conditions on the functional class.

Authors: We agree that fixed-point stability governs the decay rate of the covariance while Q is set by the lowest non-vanishing Hermite projection of the functional. Our theorems (3.1--3.3) already treat the three regimes as arising from the interaction of this decay rate with Q: slow decay yields the limiting Gaussian field functional, matched decay with Q=1 yields a Gaussian, and other combinations yield Qth chaos. For functionals annihilating lower projections (e.g., even functionals), the effective Q simply increases and the same classification applies with the adjusted Q. We have revised the abstract and added a clarifying remark in the introduction to make this dependence explicit. The existing assumptions (square-integrability and Hermite expandability) suffice; no further uniformity conditions are required, as the Diagram Formula and Stein-Malliavin bounds hold uniformly over the stated functional class. revision: partial
Referee: [Main results (presumed §3 or similar)] The derivation relies on the fixed-point structure fully determining the regimes via Hermite expansion, but without explicit verification that the mapping is one-to-one for arbitrary functionals, the central claim risks overgeneralization. A concrete counterexample or additional assumption on the functional would strengthen this.

Authors: The Hermite expansion is unique, so the leading chaos order Q is uniquely determined by any given functional. The fixed-point analysis of the covariance operator then fixes the decay rate, which selects the limit type once Q is known. To make the correspondence explicit, we have added a new paragraph in Section 3 providing concrete examples (linear, quadratic, and cubic functionals) under each fixed-point regime and verifying that the predicted limits are attained. This confirms that the regimes are determined without overgeneralization. We do not believe additional assumptions on the functional class are needed beyond those already stated, nor do we see a counterexample within the L^2 setting of the paper. revision: yes

Circularity Check

0 steps flagged

No circularity: standard limit theorems applied to independent fixed-point analysis of covariance operator

full rationale

The derivation applies classical tools (Hermite expansions, Diagram Formula, Stein-Malliavin) to the Gaussian field induced by the infinite-width network on the sphere. The fixed-point structure and stability of the iterative covariance operator are derived directly from the model definition as an independent object that controls covariance decay rates; these rates then determine which of the three regimes applies to a given functional via its chaos expansion. No equation reduces to a fitted parameter renamed as prediction, no load-bearing premise rests on self-citation, and the mapping from operator stability to limiting regime follows from the stated assumptions without tautological equivalence to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper builds on standard mathematical tools from probability theory and introduces analysis of the fixed-point structure of the covariance iteration operator as the key new element.

axioms (2)

standard math Hermite expansions and Diagram Formula are valid for the Gaussian fields involved.
Classical tools in probability for expanding functionals of Gaussians.
standard math Stein-Malliavin techniques apply to the sequences of functionals.
Standard for central and non-central limit theorems in Wiener chaos.

pith-pipeline@v0.9.0 · 5437 in / 1354 out tokens · 32251 ms · 2026-05-10T01:26:55.588340+00:00 · methodology

discussion (0)

Reference graph

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