arXiv subjects

Francesco Caravenna

Publications and source records attributed to Francesco Caravenna.

At least 19 recordsLinked to original sources

arXiv2026

The Critical 2d Stochastic Heat Flow and Related Models

In these lecture notes, we review recent progress in the study of the stochastic heat equation and its discrete analogue, the directed polymer model, in spatial dimension 2. It was discovered that a phase transition emerges on an intermediate disorder scale, with Edwards-Wilkinson (Gaussian) fluctuations in the sub-critical regime. In the critical window, a unique scaling limit has been identified and named the critical 2d stochastic heat flow. This gives a meaning to the solution of the stochastic heat equation in the critical dimension 2, which lies beyond existing solution theories for singular SPDEs. We outline the proof ideas, introduce the key ingredients, and discuss related literature on disordered systems and singular SPDEs. A list of open questions is also provided.

math.PR↗

arXiv2026

Strong Disorder for Stochastic Heat Flow and 2D Directed Polymers

The critical 2D Stochastic Heat Flow (SHF) is a universal measure-valued process that provides a notion of solution to the ill-defined 2D stochastic heat equation. We investigate the SHF in the large-time and strong-disorder regimes, proving a sharp form of local extinction: we identify the rate at which the distribution collapses to zero. We also identify the spatial scale governing the transition from vanishing mass to diverging mass, and from extinction to an averaged behavior. Corresponding results are established for the partition functions of 2D directed polymers, yielding precise free-energy estimates. Our proof provides a unified framework of change of measure and coarse-graining arguments. These results offer new insights into the 2D stochastic heat equation regularized via space-time discretization: for any regime of supercritical disorder strength $β$, including the case where $β> 0$ is kept fixed, the solution exhibits fluctuations on a superdiffusive scale.

math.PR↗

arXiv2026

From disordered systems to the Critical 2D Stochastic Heat Flow

We review our joint work on the scaling limits of disordered systems, linking the notion of disorder relevance/irrelevance to that of sub/super-criticality of singular SPDEs. This line of research culminated in the construction of the Critical 2D Stochastic Heat Flow (SHF), a universal process which provides a non-trivial solution to the Stochastic Heat Equation in dimension 2, a critical singular SPDE that lies beyond the reach of existing solution theories. The SHF also offers a rare example of a non-Gaussian scaling limit for a disordered system at its phase transition point in the critical dimension.

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

Enhanced noise sensitivity, 2D directed polymers and Stochastic Heat Flow

We investigate noise sensitivity beyond the classical setting of binary random variables, extending the celebrated result by Benjamini, Kalai, and Schramm to a wide class of functions of general random variables. Our approach yields improved bounds with optimal rates. We also consider an enhanced form of noise sensitivity which yields asymptotic independence, rather than mere decorrelation. We apply these result to establish enhanced noise sensitivity for the partition functions of 2D directed polymers, in the critical regime where they converge to the critical 2D Stochastic Heat Flow. As a consequence, we prove that the Stochastic Heat Flow is independent of the white noise arising from the disorder.

math.PR↗

arXiv2025

Singularity and regularity of the critical 2D Stochastic Heat Flow

The Critical 2D Stochastic Heat Flow (SHF) provides a natural candidate solution to the ill-posed 2D Stochastic Heat Equation with multiplicative space-time white noise. In this paper, we initiate the investigation of the spatial properties of the SHF. We prove that, as a random measure on $\mathbb{R}^2$, it is a.s. singular w.r.t. the Lebesgue measure. This is obtained by probing a "quasi-critical" regime and showing the asymptotic log-normality of the mass assigned to vanishing balls, as the disorder strength is sent to zero at a suitable rate, accompanied by similar results for critical 2D directed polymers. We also describe the regularity of the SHF, showing that it is a.s. Hölder $C^{-ε}$ for any $ε>0$, implying the absence of atoms, and we establish local convergence to zero in the long time limit.

math.PR↗

arXiv2025

Quasi-critical fluctuations for 2d directed polymers

We study the 2d directed polymer in random environment in a novel *quasi-critical regime*, which interpolates between the much studied sub-critical and critical regimes. We prove Edwards-Wilkinson fluctuations throughout the quasi-critical regime, showing that the diffusively rescaled partition functions are asymptotically Gaussian. We deduce a corresponding result for the critical 2d Stochastic Heat Flow. A key challenge is the lack of hypercontractivity, which we overcome deriving new moment estimates.

math.PR↗

arXiv2024

Hairer's multilevel Schauder estimates without Regularity Structures

We investigate the regularising properties of singular kernels at the level of germs, i.e. families of distributions indexed by points in $\mathbb{R}^d$. First we construct a suitable integration map which acts on general coherent germs. Then we focus on germs that can be decomposed along a basis (corresponding to the so-called modelled distributions in Regularity Structures) and we prove a version of Hairer's multilevel Schauder estimates in this setting, with minimal assumptions.

math.AP↗

arXiv2023

The critical 2d Stochastic Heat Flow is not a Gaussian Multiplicative Chaos

The critical $2d$ Stochastic Heat Flow (SHF) is a stochastic process of random measures on ${\mathbb R}^2$, recently constructed in [CSZ23]. We show that this process falls outside the class of Gaussian Multiplicative Chaos (GMC), in the sense that it cannot be realised as the exponential of a (generalised) Gaussian field. We achieve this by deriving strict lower bounds on the moments of the SHF that are of independent interest.

math.PR↗

arXiv2023

The Critical 2d Stochastic Heat Flow

We consider directed polymers in random environment in the critical dimension $d = 2$, focusing on the intermediate disorder regime when the model undergoes a phase transition. We prove that, at criticality, the diffusively rescaled random field of partition functions has a unique scaling limit: a universal process of random measures on $\mathbb{R}^2$ with logarithmic correlations, which we call the *Critical 2d Stochastic Heat Flow*. It is the natural candidate for the long sought solution of the critical 2d Stochastic Heat Equation with multiplicative space-time white noise.

math.PR↗

arXiv2022

Gaussian Limits for Subcritical Chaos

We present a simple criterion, only based on second moment assumptions, for the convergence of polynomial or Wiener chaos to a Gaussian limit. We exploit this criterion to obtain new Gaussian asymptotics for the partition functions of two-dimensional directed polymers in the sub-critical regime, including a singular product between the partition function and the disorder. These results can also be applied to the KPZ and Stochastic Heat Equation. As a tool of independent interest, we derive an explicit chaos expansion which sharply approximates the logarithm of the partition function.

math.PR↗

arXiv2021

Phase transitions for spatially extended pinning

We consider a directed polymer of length $N$ interacting with a linear interface. The monomers carry i.i.d. random charges $(ω_i)_{i=1}^N$ taking values in $\mathbb{R}$ with mean zero and variance one. Each monomer $i$ contributes an energy $(βω_i-h)φ(S_i)$ to the interaction Hamiltonian, where $S_i \in \mathbb{Z}$ is the height of monomer $i$ with respect to the interface, $φ: \mathbb{Z} \to [0,\infty)$ is the interaction potential, $β\in [0,\infty)$ is the inverse temperature, and $h \in \mathbb{R}$ is the charge bias parameter. The configurations of the polymer are weighted according to the Gibbs measure associated with the interaction Hamiltonian, where the reference measure is given by a Markov chain on $\mathbb{Z}$. We study both the quenched and the annealed free energy per monomer in the limit as $N\to\infty$. We show that each exhibits a phase transition along a critical curve in the $(β, h)$-plane, separating a localized phase (where the polymer stays close to the interface) from a delocalized phase (where the polymer wanders away from the interface). We derive variational formulas for the critical curves and we obtain upper and lower bounds on the quenched critical curve in terms of the annealed critical curve. In addition, for the special case where the reference measure is given by a Bessel random walk, we derive the scaling limit of the annealed free energy as $β, h \downarrow 0$ in three different regimes for the tail exponent of $φ$.

math.PR↗

arXiv2020

Hairer's Reconstruction Theorem without Regularity Structures

This survey is devoted to Martin Hairer's Reconstruction Theorem, which is one of the cornerstones of his theory of Regularity Structures. Our aim is to give a new self-contained and elementary proof of this Theorem, together with some applications, including a characterization, based on a single arbitrary test function, of negative Hölder spaces. We present the Reconstruction Theorem as a general result in the theory of distributions that can be understood without any knowledge of Regularity Structures themselves, which we do not even need to define.

math.AP↗

arXiv2019

The Dickman subordinator, renewal theorems, and disordered systems

We consider the so-called Dickman subordinator, whose Levy measure has density 1/x restricted to the interval (0,1). The marginal density of this process, known as the Dickman function, appears in many areas of mathematics, from number theory to combinatorics. In this paper, we study renewal processes in the domain of attraction of the Dickman subordinator, for which we prove local renewal theorems. We then present applications to marginally relevant disordered systems, such as pinning and directed polymer models, and prove sharp second moment estimates on their partition functions.

math.PR↗

arXiv2019

The two-dimensional KPZ equation in the entire subcritical regime

We consider the KPZ equation in space dimension 2 driven by space-time white noise. We showed in previous work that if the noise is mollified in space on scale $ε$ and its strength is scaled as $\hatβ/ \sqrt{|\log ε|}$, then a transition occurs with explicit critical point $\hatβ_c = \sqrt{2π}$. Recently Chatterjee and Dunlap showed that the solution admits subsequential scaling limits as $ε\downarrow 0$, for sufficiently small $\hatβ$. We prove here that the limit exists in the entire subcritical regime $\hatβ\in (0, \hatβ_c)$ and we identify it as the solution of an additive Stochastic Heat Equation, establishing so-called Edwards-Wilkinson fluctuations. The same result holds for the directed polymer model in random environment in space dimension 2.

math.PR↗

arXiv2019

Local large deviations and the strong renewal theorem

We establish two different, but related results for random walks in the domain of attraction of a stable law of index $α$. The first result is a local large deviation upper bound, valid for $α\in (0,1) \cup (1,2)$, which improves on the classical Gnedenko and Stone local limit theorems. The second result, valid for $α\in (0,1)$, is the derivation of necessary and sufficient conditions for the random walk to satisfy the strong renewal theorem (SRT). This solves a long standing problem, which dates back to the 1962 paper of Garsia and Lamperti [Comm. Math. Helv.] for renewal processes (i.e. random walks with non-negative increments), and to the 1968 paper of Williamson [Pacific J. Math.] for general random walks. This paper supersedes the individual preprints arXiv:1507.07502 and arXiv:1507.06790

math.PR↗

arXiv2019

On the moments of the (2+1)-dimensional directed polymer and stochastic heat equation in the critical window

The partition function of the directed polymer model on Z^{2+1} undergoes a phase transition in a suitable continuum and weak disorder limit. In this paper, we focus on a window around the critical point. Exploiting local renewal theorems, we compute the limiting third moment of the space-averaged partition function, showing that it is uniformly bounded. This implies that the rescaled partition functions, viewed as a generalized random field on R^2, have non-trivial subsequential limits, and each such limit has the same explicit covariance structure. We obtain analogous results for the stochastic heat equation on R^2, extending previous work by Bertini and Cancrini.

math.PR↗

arXiv2018

On the maximum of conditioned random walks and tightness for pinning models

We consider real random walks with finite variance. We prove an optimal integrability result for the diffusively rescaled maximum, when the walk or its bridge is conditioned to stay positive, or to avoid zero. As an application, we prove tightness under diffusive rescaling for general pinning and wetting models based on random walks.

math.PR↗