arXiv subjects

Giovanni Peccati

Publications and source records attributed to Giovanni Peccati.

At least 19 recordsLinked to original sources

arXiv2026

Correlated initialization of deep residual networks

We study the large-depth behavior of residual networks whose weights are correlated across layers at initialization. Our results confirm and extend a conjecture of Marion et al. [2025], according to which correlated initializations should interpolate continuously between the Brownian stochastic differential equation arising from independent initialization and the ordinary differential equation arising from perfectly correlated initialization. When the initialization is obtained from the application of a feature function to a stationary Gaussian sequence with regularly varying correlation, we prove that there exists a unique critical scaling such that the infinite-depth limit is the solution of a Young differential equation driven by a Hermite process. Hermite processes reduce to the fractional Brownian motion if the feature function generating the initialization has Hermite rank one, which is the case for the identity function, for example. We show that the critical scaling and asymptotic limit are uniquely determined by the decay of correlations together with the Hermite rank of the feature function. Consequently, the correlation structure and Hermite rank of the initialization represent meaningful hyperparameters in the asymptotic regime. By contrast, under finite-variance iid initialization, the asymptotic driver is universally Brownian up to normalization regardless of the choice of distribution. Our proofs rely on a collection of novel results establishing a robust stability theory for Young differential equations in Banach spaces.

math.PR↗

arXiv2026

Scars in random waves and the FGF 1/2 universality class

We study the large-domain asymptotics of geometric observables in Berry's random wave model on $\mathbb{R}^d$. We show that, in sharp contrast with the behavior of stationary random fields with absolutely continuous spectral measures, any observable whose fluctuations are asymptotically fully correlated with its second Wiener chaos projection belongs to a common universality class governed by a fractional Gaussian field with Hurst index $H=(1-d)/2$. This class also includes the classical stationary Poisson line process in $\mathbb{R}^d$. Our findings show that suitable raw observables of Berry's random wave (such as critical point counts or non-nodal level set volumes) have large-domain fluctuations that become arbitrarily close -- in the sense of random tempered distributions -- to those generated by a (possibly noisy) Poisson line process. This probabilistic approximation provides evidence that the large-scale filamentary patterns observed in numerical simulations of random waves -- often referred to as "scars" or "scarlets" following the numerical investigations of Heller, O'Connor and Gehlen (1987)-- may admit a natural probabilistic interpretation. In the second part of our work, we characterize the scaling limit -- in a distributional sense -- of suitable quadratic transformations of the Radon--Fourier coefficients associated with a large class of stationary fields. We show that random waves are characterized by the property that such a scaling limit is a generalized random field obtained by composing white noise on the affine Grassmannian of lines with a dimension-dependent deterministic operator. As an application of our main results, we derive explicit conditions ensuring that quadratic functionals of pullback monochromatic waves on compact Riemannian manifolds exhibit distributional limits in the fractional Gaussian universality class described above.

math.PR↗

arXiv2025

Tightness of Stationary Nodal Measures

We study the rescaled nodal volume field $\xi_R$ associated with a smooth, stationary Gaussian field on $[0,R]^d$, whose covariance satisfies adequate integrability conditions. Our main theorem shows that, as $R \to \infty$, the process $\xi_R$ converges in distribution, in an appropriate space of c\`adl\`ag mappings, to a standard Brownian sheet. The proof relies on a recent finite-dimensional CLT by Ancona, Gass, Letendre, and Stecconi (2025), as well as on a multidimensional Kolmogorov--Chentsov criterion for tightness due to Bickel and Wichura (1971). The application of the latter requires new moment estimates that are of independent interest. Our results stand in sharp contrast with Berry's random wave model, where the required integrability conditions fail and the question of tightness remains open.

math.PR↗

arXiv2025

Universal Cancellations in Uniform Random Waves

A vast literature over the past fifteen years has been devoted to the study of the geometric properties of Gaussian random waves. In this work, we investigate the geometric behavior of \emph{uniform random waves}, a much less studied non-Gaussian model in which the $L^2$ norm is constrained to be exactly equal to one in every realization (a normalization that is natural from the standpoint of quantum mechanics). We show that this norm-constrained formulation has deep consequences for the universality of the so-called \emph{Berry's cancellation phenomenon}, as well as for novel high-frequency asymptotic variance estimates. These effects manifest themselves in both local geometric functionals, such as the Lipschitz--Killing curvatures, and global ones, such as the number of connected components above a fixed threshold. A key byproduct of our analysis is a new explicit relation between Hermite expansions and spherical harmonic decompositions for $0$-homogeneous functionals of Gaussian vectors, which enables a systematic chaos-based analysis of non-Gaussian random waves.

math.PR↗

arXiv2025

Optimal local central limit theorems on Wiener chaos

This paper investigates a local central limit theorem for a normalized sequence of random variables belonging to a fixed order Wiener chaos and converging to the standard normal distribution. We prove, without imposing any additional conditions, that the optimal rate of convergence of their density functions to the standard normal density in the Sobolev space $W^{k,r}(\mathbb{R})$, for every $k \in \mathbb{N} \cup \{0\}$ and $r \in [1,\infty]$, is determined by the maximum of the absolute values of their third and fourth cumulants. We also obtain exact asymptotics for this convergence under an additional assumption. Our approach is based on Malliavin--Stein techniques combined with tools from the theory of generalized functionals in Malliavin calculus.

math.PR↗

arXiv2025

On irreducible central limit theorems

We consider sequences of homogeneous sums based on independent random variables and satisfying a central limit theorem (CLT). We address the following question: "In which cases is it not possible to reduce such an asymptotic result to the classical Lindeberg-Feller CLT through a restriction of the summation domain?". We provide several sufficient conditions for such irreducibility, expressed both in terms of (hyper)graphs Laplace eigenvalues, and of a certain notion of combinatorial dimension. Our analysis combines Cheeger-type inequalities with fourth moment theorems, showing that the irreducibility of a given CLT for homogeneous sums can be naturally encoded by the connectivity properties of the associated sequence of weighted hypergraphs. Several ad-hoc constructions are provided in the special case of quadratic forms.

math.PR↗

arXiv2025

A Caveat on Metrizing Convergence in Distribution on Hilbert Spaces

We consider Sobolev-type distances on probability measures over separable Hilbert spaces involving the Schatten-$p$ norms, which include as special cases a distance first introduced by Bourguin and Campese (2020) when $p=2$, and a distance introduced by Gin\'e and Leon (1980) when $p=\infty$. Our analysis shows that, unless $p=\infty$, these distances fail to metrize convergence in distribution in infinite dimensions. This clarifies several inconsistencies and misconceptions in the recent literature that arose from confusion between different types of distances.

math.PR↗

arXiv2025

Power-law correction in the probability density function of the critical Ising magnetization

At the critical point, the probability density function of the Ising magnetization is believed to decay like $\exp{(-x^{\delta+1})}$, where $\delta$ is the Ising critical exponent that controls the decay to zero of the magnetization in a vanishing external field. In this paper, we discuss the presence of a power-law correction $x^{\frac{\delta-1}{2}}$, which has been debated in the physics literature. We argue that whether such a correction is present or not is related to the asymptotic behavior of a function that measures the extent to which the average magnetization of a finite system with an external field is influenced by the boundary conditions. Our discussion is informed by a mixture of heuristic calculations and rigorous results. Along the way, we review some recent results on the critical Ising model and prove properties of the average magnetization in two dimensions which are of independent interest.

cond-mat.stat-mech↗

arXiv2025

The Poisson Multiplication Formula

We establish necessary and sufficient conditions implying that the product of $m\geq 2$ Poisson functionals, living in a finite sum of Wiener chaoses, is square-integrable. Our conditions are expressed in terms of iterated add-one cost operators, and are obtained through the use of a novel family of Poincar\'e inequalities for almost surely finite random variables, generalizing the recent findings by Trauthwein (2024). When specialized to the case of multiple Wiener-It\^o integrals, our results yield general multiplication formulae on the Poisson space under minimal conditions, naturally expressed in terms of partitions and diagrams. Our work addresses several questions left open in a seminal work by Surgailis (1984), and completes a line of research initiated in D\"obler and Peccati (2018).

math.PR↗

arXiv2025

Entropic bounds for conditionally Gaussian vectors and applications to neural networks

Using entropic inequalities from information theory, we provide new bounds on the total variation and 2-Wasserstein distances between a conditionally Gaussian law and a Gaussian law with invertible covariance matrix. We apply our results to quantify the speed of convergence to Gaussian of a randomly initialized fully connected neural network and its derivatives - evaluated in a finite number of inputs - when the initialization is Gaussian and the sizes of the inner layers diverge to infinity. Our results require mild assumptions on the activation function, and allow one to recover optimal rates of convergence in a variety of distances, thus improving and extending the findings of Basteri and Trevisan (2023), Favaro et al. (2023), Trevisan (2024) and Apollonio et al. (2024). One of our main tools are the quantitative cumulant estimates established in Hanin (2024). As an illustration, we apply our results to bound the total variation distance between the Bayesian posterior law of the neural network and its derivatives, and the posterior law of the corresponding Gaussian limit: this yields quantitative versions of a posterior CLT by Hron et al. (2022), and extends several estimates by Trevisan (2024) to the total variation metric.

math.PR↗

arXiv2024

Spectra of Poisson functionals and applications in continuum percolation

Let $η$ be a Poisson random measure (defined on some Polish space), and let $F(η)$ be a square-integrable functional of $η$. In this paper we define and study a new notion of {\it spectral point process} associated with $F(η)$, and use such an object to study sharp noise instability and sensitivity properties of planar critical continuum percolation models under spatial birth-death (Ornstein-Uhlenbeck) dynamics -- the notion of sharp noise instability being a natural strengthening of the absence of noise stability. The concept of spectral point process is defined by exploiting the Wiener-Itô chaos expansion of $F$, and represents a natural continuum counterpart to the notion of {\it spectral sample}, as introduced in Garban, Pete and Schramm (2010), in the context of discrete percolation models. In the particular case where $η$ is a marked Poisson measure, we use Hoeffding-ANOVA decompositions to establish an explicit connection with the notion of {\it annealed spectral sample}, introduced in Vanneuville (2021) in the context of Poisson-Voronoi percolation. We also relate spectral processes with an appropriate notion of {\it pivotal processes}. As applications, we show sharp noise instability of crossing events in the critical Poisson Boolean model with unit-radius balls and, using an observation of Vanneuville, we obtain sharp noise sensitivity (as well as sharp noise instability) for crossing events in the Poisson Voronoi percolation model. As an important ingredient, we prove quasi-multiplicativity of the $4$-arm probabilities in the critical Poisson Boolean percolation model.

math.PR↗

arXiv2024

Quantitative CLTs in Deep Neural Networks

We study the distribution of a fully connected neural network with random Gaussian weights and biases in which the hidden layer widths are proportional to a large constant $n$. Under mild assumptions on the non-linearity, we obtain quantitative bounds on normal approximations valid at large but finite $n$ and any fixed network depth. Our theorems show both for the finite-dimensional distributions and the entire process, that the distance between a random fully connected network (and its derivatives) to the corresponding infinite width Gaussian process scales like $n^{-γ}$ for $γ>0$, with the exponent depending on the metric used to measure discrepancy. Our bounds are strictly stronger in terms of their dependence on network width than any previously available in the literature; in the one-dimensional case, we also prove that they are optimal, i.e., we establish matching lower bounds.

cs.LG↗

arXiv2024

Nodal Volumes as Differentiable Functionals of Gaussian fields

We characterize the absolute continuity of the law and the Malliavin-Sobolev regularity of random nodal volumes associated with smooth Gaussian fields on generic $\mathcal{C}^2$ manifolds with arbitrary dimension. Our results extend and generalize the seminal contribution by Angst and Poly (2020) about stationary fields on Euclidean spaces and cover, in particular, the case of two-dimensional manifolds, possibly with boundary and corners. The main tools exploited in the proofs include the use of Gaussian measures on Banach spaces, Morse theory, and the characterization of Malliavin-Sobolev spaces in terms of ray absolute continuity. Several examples are analyzed in detail.

math.PR↗

arXiv2023

Nonlinear Functionals of Hyperbolic Random Waves: the Wiener Chaos Approach

We consider Gaussian random waves on hyperbolic spaces and establish variance asymptotics and central limit theorems for a large class of their integral functionals, both in the high-frequency and large domain limits. Our strategy of proof relies on a fine analysis of Wiener chaos expansions, which in turn requires us to analytically assess the fluctuations of integrals involving mixed moments of covariance kernels. Our results complement several recent findings on non-linear transforms of planar and arithmetic random waves, as well as of random spherical harmonics. In the particular case of 2-dimensional hyperbolic spaces, our analysis reveals an intriguing discrepancy between the high-frequency and large domain fluctuations of the so-called fourth polyspectra -- a phenomenon that has no counterpart in the Euclidean setting. We develop applications of a geometric flavor, most notably to excursion volumes and occupation densities.

math.PR↗

arXiv2023

Nodal Statistics of Planar Random Waves

We consider Berry's random planar wave model (1977) for a positive Laplace eigenvalue $E>0$, both in the real and complex case, and prove limit theorems for the nodal statistics associated with a smooth compact domain, in the high-energy limit ($E\to \infty$). Our main result is that both the nodal length (real case) and the number of nodal intersections (complex case) verify a Central Limit Theorem, which is in sharp contrast with the non-Gaussian behaviour observed for real and complex arithmetic random waves on the flat $2$-torus, see Marinucci et al. (2016) and Dalmao et al. (2016). Our findings can be naturally reformulated in terms of the nodal statistics of a single random wave restricted to a compact domain diverging to the whole plane. As such, they can be fruitfully combined with the recent results by Canzani and Hanin (2016), in order to show that, at any point of isotropic scaling and for energy levels diverging sufficently fast, the nodal length of any Gaussian pullback monochromatic wave verifies a central limit theorem with the same scaling as Berry's model. As a remarkable byproduct of our analysis, we rigorously confirm the asymptotic behaviour for the variances of the nodal length and of the number of nodal intersections of isotropic random waves, as derived in Berry (2002).

math.PR↗

arXiv2023

Phase Singularities in Complex Arithmetic Random Waves

Complex arithmetic random waves are stationary Gaussian complex-valued solutions of the Helmholtz equation on the two-dimensional flat torus. We use Wiener-Itô chaotic expansions in order to derive a complete characterization of the second order high-energy behaviour of the total number of phase singularities of these functions. Our main result is that, while such random quantities verify a universal law of large numbers, they also exhibit non-universal and non-central second order fluctuations that are dictated by the arithmetic nature of the underlying spectral measures. Such fluctuations are qualitatively consistent with the cancellation phenomena predicted by Berry (2002) in the case of complex random waves on compact planar domains. Our results extend to the complex setting recent pathbreaking findings by Rudnick and Wigman (2008), Krishnapur, Kurlberg and Wigman (2013) and Marinucci, Peccati, Rossi and Wigman (2016). The exact asymptotic characterization of the variance is based on a fine analysis of the Kac-Rice kernel around the origin, as well as on a novel use of combinatorial moment formulae for controlling long-range weak correlations.

math.PR↗

arXiv2022

Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees

Proofs of sharp phase transition and noise sensitivity in percolation have been significantly simplified by the use of randomized algorithms, via the OSSS inequality (proved by O'Donnell, Saks, Schramm and Servedio (2005)) and the Schramm-Steif inequality for the Fourier-Walsh coefficients of functions defined on the Boolean hypercube. In this article, we prove intrinsic versions of the OSSS and Schramm-Steif inequalities for functionals of a general Poisson process, and apply these new estimates to deduce sufficient conditions - expressed in terms of randomized stopping sets - yielding sharp phase transitions, quantitative noise sensitivity, exceptional times and bounds on critical windows for monotonic Boolean Poisson functions. Our analysis is based on a new general definition of `stopping set', not requiring any topological property for the underlying measurable space, as well as on the new concept of a `continuous-time decision tree', for which we establish several fundamental properties. We apply our findings to the $k$-percolation of the Poisson Boolean model and to the Poisson-based confetti percolation with bounded random grains. In these two models, we reduce the proof of sharp phase transitions for percolation, and of noise sensitivity for crossing events, to the construction of suitable randomized stopping sets and the computation of one-arm probabilities. This enables us to settle some open problem suggested by Ahlberg, Tassion and Texeira (2018) on noise sensitivity of crossing events for the planar Poisson Boolean model and also planar Confetti percolation model. Further, we also prove that critical probability is $1/2$ in certain planar confetti percolation models. A special case of this result was conjectured by Benjamini and Schramm (1998) and proved by Müller (2017). Other special cases were proven by Hirsch (2015) and Ghosh and Roy (2018).

math.PR↗

arXiv2022

Functional Convergence of Berry's Nodal Lengths: Approximate Tightness and Total Disorder

We consider Berry's random planar wave model (1977), and prove spatial functional limit theorems - in the high-energy limit - for discretized and truncated versions of the random field obtained by restricting its nodal length to rectangular domains. Our analysis is crucially based on a detailed study of the projection of nodal lengths onto the so-called second Wiener chaos, whose high-energy fluctuations are given by a Gaussian total disorder field indexed by polygonal curves. Such an exact characterization is then combined with moment estimates for suprema of stationary Gaussian random fields, and with a tightness criterion by Davydov and Zikitis (2005).

math.PR↗