Phase Transitions in the Fluctuations of Functionals of Random Neural Networks
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913057620361216 |
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| author | Di Lillo, Simmaco Maini, Leonardo Marinucci, Domenico |
| author_facet | Di Lillo, Simmaco Maini, Leonardo Marinucci, Domenico |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19738 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Phase Transitions in the Fluctuations of Functionals of Random Neural Networks Di Lillo, Simmaco Maini, Leonardo Marinucci, Domenico Probability Machine Learning 60G60, 68T07 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. |
| title | Phase Transitions in the Fluctuations of Functionals of Random Neural Networks |
| topic | Probability Machine Learning 60G60, 68T07 |
| url | https://arxiv.org/abs/2604.19738 |