arXiv subjects

Giuseppe Cannizzaro

Publications and source records attributed to Giuseppe Cannizzaro.

At least 19 recordsLinked to original sources

arXiv2026

Equilibrium fluctuations for a multi-species particle system with long jumps

In the present paper, we study the equilibrium fluctuations of a particle system in infinite volume with two conserved quantities and long-range dependence. More specifically, the model of interest is the so-called ABC model, in which three types of particles (A, B and C) exchange their locations between $x\in\mathbb{Z}$ and $x+z\in\mathbb{Z}$ at a rate that depends on the type of particles involved and is proportional to $|z|^{-γ-1}$ for $γ>0$. After rigorously identifying the normal modes associated to the conserved quantities (the density of particles of types $A$ and $B$, say), we prove that their fluctuations converge to independent fractional stochastic partial differential equations (SPDEs), which are either Gaussian or the Stochastic Burgers equation, and whose nature is determined by the microscopic range of dependence and the strength of the asymmetry.

math.PR↗

arXiv2026

Superdiffusive central limit theorem for a class of driven diffusive systems at the critical dimension

We study the large-scale behaviour of a class of driven diffusive systems modelled by a Stochastic Partial Differential Equation, the Stochastic Burgers Equation (SBE) with general nonlinearity, at the critical dimension and in infinite volume. Our main result shows that, under a logarithmically superdiffusive space-time scaling, it is given by the same explicit Gaussian Fixed point obtained in [G. Cannizzaro, Q. Moulard, & F. Toninelli, arxiv.org/abs/2501.00344, 2025] for the quadratic SBE, but with suitably renormalised coefficients, thereby rigorously justifying and partly correcting the classical Physics derivation of the SBE in [H. van Beijeren, R. Kutner, & H. Spohn, Phys. Rev. Lett., 1986] based on Spohn's theory of nonlinear fluctuating hydrodynamics. Besides, ours is the first universality-type result for out-of-equilibrium systems and the first extension of [M. Hairer, J. Quastel, Forum of Mathematics, Pi, Vol. 6, 2018, e3], to the critical dimension and beyond weak coupling. The major challenge in our work is the mild growth condition on the nonlinearity which renders even the well-posedness of the microscopic equation non-trivial. Additional key novelties include the derivation of fine estimates on the non-quadratic part of the generator as well as a new approximation for the resolvent associated to the solution of the quadratic SBE.

math.PR↗

arXiv2025

Top of the spectrum of discrete Anderson Hamiltonians with correlated Gaussian potentials

We investigate the top of the spectrum of discrete Anderson Hamiltonians with correlated Gaussian noise in the large volume limit. The class of Gaussian noises under consideration allows for long-range correlations. We show that the largest eigenvalues converge to a Poisson point process and we obtain a very precise description of the associated eigenfunctions near their localisation centres. We also relate these localisation centres with the locations of the maxima of the noise. Actually, our analysis reveals that this relationship depends in a subtle way on the behaviour near $0$ of the covariance function of the noise: in some situations, the largest eigenfunctions are not associated with the largest values of the noise.

math.PR↗

arXiv2025

Superdiffusive Central Limit Theorem for the Stochastic Burgers Equation at the critical dimension

The Stochastic Burgers Equation (SBE) is a singular, non-linear Stochastic Partial Differential Equation (SPDE) that describes, on mesoscopic scales, the fluctuations of stochastic driven diffusive systems with a conserved scalar quantity. In space dimension d = 2, the SBE is critical, being formally scale invariant under diffusive scaling. As such, it falls outside of the domain of applicability of the theories of Regularity Structures and paracontrolled calculus. In apparent contrast with the formal scale invariance, we fully prove the conjecture first appeared in [H. van Beijeren, R. Kutner, & H. Spohn, Phys. Rev. Lett., 1986] according to which the 2d-SBE is logarithmically superdiffusive, i.e. its diffusion coefficient diverges like $(\log t)^{2/3}$ as $t\to\infty$, thus removing subleading diverging multiplicative corrections in [D. De Gaspari & L. Haunschmid-Sibitz, Electron. J. Probab., 2024] and in [H.-T. Yau, Ann. of Math., 2004] for 2d-ASEP. We precisely identify the constant prefactor of the logarithm and show it is proportional to $λ^{4/3}$, for $λ>0$ the coupling constant, which, intriguingly, turns out to be exactly the same as for the one-dimensional Stochastic Burgers/KPZ equation. More importantly, we prove that, under super-diffusive space-time rescaling, the SBE has an explicit Gaussian fixed point in the Renormalization Group sense, by deriving a superdiffusive central limit-type theorem for its solution. This is the first scaling limit result for a critical singular SPDE, beyond the weak coupling regime, and is obtained via a refined control, on all length-scales, of the resolvent of the generator of the SBE. We believe our methods are well-suited to study other out-of-equilibrium driven diffusive systems at the critical dimension, such as 2d-ASEP, which, we conjecture, have the same large-scale Fixed Point as SBE.

math.PR↗

arXiv2024

From ABC to KPZ

We study the equilibrium fluctuations of an interacting particle system evolving on the discrete ring with $N\in\mathbb N$ points, denoted by $\mathbb T_N$, and with three species of particles that we name $A,B$ and $C$, but such that at each site there is only one particle. We prove that proper choices of density fluctuation fields (that match those from nonlinear fluctuating hydrodynamics theory) associated to the (two) conserved quantities converge, in the limit $N\to\infty$, to a system of stochastic partial differential equations, that can either be the Ornstein-Uhlenbeck equation or the Stochastic Burgers equation. To understand the cross interaction between the two conserved quantities, we derive a general version of the Riemann-Lebesgue lemma which is of independent interest.

math.PR↗

arXiv2024

Lecture notes on stationary critical and super-critical SPDEs

The goal of these lecture notes is to present recent results regarding the large-scale behaviour of critical and super-critical non-linear stochastic PDEs, that fall outside the realm of the theory of Regularity Structures. These include the two-dimensional Anisotropic KPZ equation, the stochastic Burgers equation in dimension $d\ge 2$ and the stochastic Navier-Stokes equation with divergence-free noise in dimension $d=2$. Rather than providing complete proofs, we try to emphasise the main ideas, and some crucial aspects of our approach: the role of the generator equation and of the Fluctuation-Dissipation Theorem to identify the limit process; Wiener chaos decomposition with respect to the stationary measure and its truncation; and the so-called Replacement Lemma, which controls the weak coupling limit of the equations in the critical dimension and identifies the limiting diffusivity. For pedagogical reasons, we will focus exclusively on the stochastic Burgers equation. The notes are based on works in collaboration with Dirk Erhard and Massimiliano Gubinelli.

math.PR↗

arXiv2024

An invariance principle for the 2d weakly self-repelling Brownian polymer

We investigate the large-scale behaviour of the Self-Repelling Brownian Polymer (SRBP) in the critical dimension $d=2$. The SRBP is a model of self-repelling motion, which is formally given by the solution a stochastic differential equation driven by a standard Brownian motion and with a drift given by the negative gradient of its own local time. As with its discrete counterpart, the "true" self-avoiding walk (TSAW) of [D.J. Amit, G. Parisi, & L. Peliti, Asymptotic behaviour of the "true" self-avoiding walk, Phys. Rev. B, 1983], it is conjectured to be logarithmically superdiffusive, i.e. to be such that its mean-square displacement grows as $t(\log t)^β$ for $t$ large and some currently unknown $β\in(0,1)$. The main result of the paper is an invariance principle for the SRBP under the weak coupling scaling, which corresponds to scaling the SRBP diffusively and simultaneously tuning down the strength of the self-interaction in a scale-dependent way. The diffusivity for the limiting Brownian motion is explicit and its expression provides compelling evidence that the $β$ above should be $1/2$. Further, we derive the scaling limit of the so-called environment seen by the particle process, which formally solves a non-linear singular stochastic PDE of transport-type, and prove this is given by the solution of a stochastic linear transport equation with enhanced diffusivity.

math.PR↗

arXiv2023

Gaussian Fluctuations for the stochastic Burgers equation in dimension $d\geq 2$

The goal of the present paper is to establish a framework which allows to rigorously determine the large-scale Gaussian fluctuations for a class of singular SPDEs at and above criticality, and therefore beyond the range of applicability of pathwise techniques, such as the theory of Regularity Structures. To this purpose, we focus on a $d$-dimensional generalization of the Stochastic Burgers equation (SBE) introduced in [H. van Beijeren, R. Kutner and H. Spohn, Excess noise for driven diffusive systems, PRL, 1985]. In both the critical $d=2$ and super-critical $d\geq 3$ cases, we show that the scaling limit of (the regularised) SBE is given by a stochastic heat equation with non-trivially renormalised coefficient, introducing a set of tools that we expect to be applicable more widely. For $d\ge3$ the scaling adopted is the classical diffusive one, while in $d=2$ it is the weak coupling scaling which corresponds to tuning down the strength of the interaction in a scale-dependent way.

math.PR↗

arXiv2023

The Brownian Web as a random $\mathbb R$-tree

Motivated by [G. Cannizzaro, M. Hairer, Comm. Pure Applied Math., '22], we provide a construction of the Brownian Web (see [Tóth B., Werner W., Probab. Theory Related Fields, '98] and [L. R. G. Fontes, M. Isopi, C. M. Newman, and K. Ravishankar, Ann. Probab., '04]), i.e. a family of coalescing Brownian motions starting from every point in $\mathbb R^2$, as a random variable taking values in the space of (spatial) $\mathbb R$-trees. This gives a stronger topology than the classical one {(i.e.\ Hausdorff convergence on closed sets of paths)}, thus providing us with more continuous functions of the Brownian Web and ruling out a number of potential pathological behaviours. Along the way, we introduce a modification of the topology of spatial $\mathbb R$-trees in [T. Duquesne, J.-F. Le Gall, Probab. Theory Related Fields, '05] and [M. T. Barlow, D. A. Croydon, T. Kumagai, Ann. Probab. '17] which makes it a complete separable metric space and could be of independent interest. We determine some properties of the characterisation of the Brownian Web in this context (e.g.\ its box-counting dimension) and recover some which were determined in earlier works, such as duality, special points and convergence of the graphical representation of coalescing random walks.

math.PR↗

arXiv2022

$\sqrt{\log t}$-superdiffusivity for a Brownian particle in the curl of the 2d GFF

The present work is devoted to the study of the large time behaviour of a critical Brownian diffusion in two dimensions, whose drift is divergence-free, ergodic and given by the curl of the 2-dimensional Gaussian Free Field. We prove the conjecture, made in [B. Tóth, B. Valkó, J. Stat. Phys., 2012], according to which the diffusion coefficient $D(t)$ diverges as $\sqrt{\log t}$ for $t\to\infty$. Starting from the fundamental work by Alder and Wainwright [B. Alder, T. Wainright, Phys. Rev. Lett. 1967], logarithmically superdiffusive behaviour has been predicted to occur for a wide variety of out-of-equilibrium systems in the critical spatial dimension $d=2$. Examples include the diffusion of a tracer particle in a fluid, self-repelling polymers and random walks, Brownian particles in divergence-free random environments, and, more recently, the 2-dimensional critical Anisotropic KPZ equation. Even if in all of these cases it is expected that $D(t)\sim\sqrt{\log t}$, to the best of the authors' knowledge, this is the first instance in which such precise asymptotics is rigorously established.

math.PR↗

arXiv2021

Stationary stochastic Navier-Stokes on the plane at and above criticality

In the present paper, we study the fractional incompressible Stochastic Navier-Stokes equation on $\mathbb{R}^2$, formally defined as \[ \partial_t v = -\tfrac12 (-Δ)^θv - λv \cdot \nabla v + \nabla p - \nabla^{\perp} (-Δ)^{\frac{θ-1}{2}} ξ, \qquad \nabla \cdot v = 0 \, , \] where $θ\in(0,1]$, $ξ$ is the space-time white noise on $\mathbb{R}_+\times\mathbb{R}^2$ and $λ$ is the coupling constant. For any value of $θ$ the previous equation is ill-posed due to the singularity of the noise, and is critical for $θ=1$ and supercritical for $θ\in(0,1)$. For $θ=1$, we prove that the weak coupling regime for the equation, i.e. regularisation at scale $N$ and coupling constant $λ=\hatλ/\sqrt{\log N}$, is meaningful in that the sequence $\{v^N\}_N$ of regularised solutions is tight and the nonlinearity does not vanish as $N\to\infty$. Instead, for $θ\in(0,1)$ we show that the large scale behaviour of $v$ is trivial, as the nonlinearity vanishes and $v$ is simply converges to the solution of the original equation but with $λ=0$.

math.PR↗

arXiv2021

The stationary AKPZ equation: logarithmic superdiffusivity

We study the two-dimensional Anisotropic KPZ equation (AKPZ) formally given by \begin{equation*} \partial_t H=\frac12ΔH+λ((\partial_1 H)^2-(\partial_2 H)^2)+ξ\,, \end{equation*} where $ξ$ is a space-time white noise and $λ$ is a strictly positive constant. While the classical two-dimensional KPZ equation, whose nonlinearity is $|\nabla H|^2=(\partial_1 H)^2+(\partial_2 H)^2$, can be linearised via the Cole-Hopf transformation, this is not the case for AKPZ. We prove that the stationary solution to AKPZ (whose invariant measure is the Gaussian Free Field) is superdiffusive: its diffusion coefficient diverges for large times as $\sqrt{\log t}$ up to $\log\log t$ corrections, in a Tauberian sense. Morally, this says that the correlation length grows with time like $t^{1/2}\times (\log t)^{1/4}$. Moreover, we show that if the process is rescaled diffusively ($t\to t/\varepsilon^2, x\to x/\varepsilon, \varepsilon\to0$), then it evolves non-trivially already on time-scales of order approximately $1/\sqrt{|\log\varepsilon|}\ll1$. Both claims hold as soon as the coefficient $λ$ of the nonlinearity is non-zero. These results are in contrast with the belief, common in the mathematics community, that the AKPZ equation is diffusive at large scales and, under simple diffusive scaling, converges the two-dimensional Stochastic Heat Equation (2dSHE) with additive noise (i.e. the case $λ=0$).

math.PR↗

arXiv2021

Weak coupling limit of the Anisotropic KPZ equation

In the present work, we study the two-dimensional anisotropic KPZ equation (AKPZ), which is formally given by \begin{equation*} \partial_t h=\tfrac12 Δh + λ((\partial_1 h)^2)-(\partial_2 h)^2) +ξ\,, \end{equation*} where $ξ$ denotes a space-time white noise and $λ>0$ is the so-called coupling constant. The AKPZ equation is a {\it critical} SPDE, meaning that not only it is analytically ill-posed but also the breakthrough path-wise techniques for singular SPDEs [M. Hairer, Ann. Math. 2014] and [M. Gubinelli, P. Imkeller and N. Perkowski, Forum of Math., Pi, 2015] are not applicable. As shown in [G. Cannizzaro, D. Erhard, F. Toninelli, arXiv, 2020], the equation regularised at scale $N$ has a diffusion coefficient that diverges logarithmically as the regularisation is removed in the limit $N\to\infty$. Here, we study the \emph{weak coupling limit} where $λ=λ_N=\hatλ/\sqrt{\log N}$: this is the correct scaling that guarantees that the nonlinearity has a still non-trivial but non-divergent effect. In fact, as $N\to\infty$ the sequence of equations converges to the linear stochastic heat equation \begin{equation*} \partial_t h =\tfrac{ν_{\rm eff}}{2} Δh + \sqrt{ν_{\rm eff}}ξ\,, \end{equation*} where $ν_{\rm eff} >1$ is explicit and depends non-trivially on $\hatλ$. This is the first full renormalization-type result for a critical, singular SPDE which cannot be linearised via Cole-Hopf or any other transformation.

math.PR↗

arXiv2021

The Brownian Castle

We introduce a $1+1$-dimensional temperature-dependent model such that the classical ballistic deposition model is recovered as its zero-temperature limit. Its $\infty$-temperature version, which we refer to as the $0$-Ballistic Deposition ($0$-BD) model, is a randomly evolving interface which, surprisingly enough, does {\it not} belong to either the Edwards--Wilkinson (EW) or the Kardar--Parisi--Zhang (KPZ) universality class. We show that $0$-BD has a scaling limit, a new stochastic process that we call {\it Brownian Castle} (BC) which, although it is "free", is distinct from EW and, like any other renormalisation fixed point, is scale-invariant, in this case under the $1:1:2$ scaling (as opposed to $1:2:3$ for KPZ and $1:2:4$ for EW). In the present article, we not only derive its finite-dimensional distributions, but also provide a "global" construction of the Brownian Castle which has the advantage of highlighting the fact that it admits backward characteristics given by the (backward) Brownian Web (see [Tóth B., Werner W., Probab. Theory Related Fields, '98] and [L. R. G. Fontes, M. Isopi, C. M. Newman, and K. Ravishankar, Ann. Probab., '04]). Among others, this characterisation enables us to establish fine pathwise properties of BC and to relate these to special points of the Web. We prove that the Brownian Castle is a (strong) Markov and Feller process on a suitable space of càdlàg functions and determine its long-time behaviour. At last, we give a glimpse to its universality by proving the convergence of $0$-BD to BC in a rather strong sense.

math.PR↗

arXiv2020

Logarithmic superdiffusivity of the 2-dimensional anisotropic KPZ equation

We study an anisotropic variant of the two-dimensional Kardar-Parisi-Zhang equation, that is relevant to describe growth of vicinal surfaces and has Gaussian, logarithmically rough, stationary states. While the folklore belief (based on one-loop Renormalization Group) is that the equation has the same scaling behaviour as the (linear) Edwards-Wilkinson equation, we prove that, on the contrary, the non-linearity induces the emergence of a logarithmic super-diffusivity. This phenomenon is similar in flavour to the super-diffusivity for two-dimensional fluids and driven particle systems.

cond-mat.stat-mech↗

arXiv2018

Malliavin Calculus for regularity structures: the case of gPAM

Malliavin calculus is implemented in the context of [M. Hairer, A theory of regularity structures, Invent. Math. 2014]. This involves some constructions of independent interest, notably an extension of the structure which accomodates a robust, and purely deterministic, translation operator, in $L^2$-directions, between "models". In the concrete context of the generalized parabolic Anderson model in 2D - one of the singular SPDEs discussed in the afore-mentioned article - we establish existence of a density at positive times.

math.PR↗

arXiv2018

Space-time discrete KPZ equation

We study a general family of space-time discretizations of the KPZ equation and show that they converge to its solution. The approach we follow makes use of basic elements of the theory of regularity structures [M. Hairer, A theory of regularity structures, Invent. Math. 2014] as well as its discrete counterpart [M. Hairer, K. Matetski, Discretizations of rough stochastic PDEs, 2015]. Since the discretization is in both space and time and we allow non-standard discretization for the product, the methods mentioned above have to be suitably modified in order to accommodate the structure of the models under study.

math.PR↗

arXiv2017

Multidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potential

We study existence and uniqueness of solution for stochastic differential equations with distributional drift by giving a meaning to the Stroock-Varadhan martingale problem associated such equations. The approach we exploit is the one of paracontrolled distributions introduced in [13]. As a result we make sense of the three dimensional polymer measure with white noise potential.

math.PR↗