arXiv subjects

Leandro Chiarini

Publications and source records attributed to Leandro Chiarini.

8 recordsLinked to original sources

arXiv2026

Second-order fields for stochastic partial differential equations

In this article, we study the second-order fluctuation of the solutions of one-dimensional polynomial stochastic partial differential equations (SPDEs) of the form \begin{equation*} (\partial_t - \Delta) \Phi_{\varepsilon} = -P(\Phi_{\varepsilon}) + \xi_{\varepsilon}, \end{equation*} where $P$ is a polynomial of degree greater than or equal to $2$, $\xi_\varepsilon$ is the white-noise after being convoluted (in space) by the heat kernel $K_\varepsilon = e^{\varepsilon \Delta}$. More precisely, taking advantage of the local solutions of pointwise well-posedness of limits $\Phi= \lim_{\varepsilon \to 0}\Phi_{\varepsilon}$, we characterise the limit of $\Phi^{err}=\lim_{\varepsilon \to 0} \varepsilon^{-1}(\Phi_{\varepsilon}-\Phi)$ as the solution of a more irregular stochastic partial differential equation. This is performed by applying the Da Prato--Debussche decomposition in the non-linear equation and characterising the limit of each of the terms. We also discuss possible characterisations of second-order fluctuations performed for higher-dimensional SPDEs in the weakly-coupled regime.

math.PR↗

arXiv2026

Imry-Ma phenomenon for the hard-core model on $\mathbb{Z}^{2}$

The \emph{Imry-Ma phenomenon} refers to the dramatic effect that disorder can have on first-order phase transitions for two-dimensional spin systems. The most famous example is the absence of a phase transition for the two-dimensional random-field Ising model. This paper establishes that a similar phenomena takes place for the hard-core model, a discrete model of crystallization: arbitrarily weak disorder prevents the formation of a crystal. Our proof of this behaviour is an adaptation of the Aizenman-Wehr argument for the Imry-Ma phenomenon, with the use of internal (spin space) symmetries for spin systems being replaced by the use spatial symmetries.

math.PR↗

arXiv2025

Sharp barrier estimates for Bessel bridges

In this article, we derive precise estimates for the probability that a Bessel bridge of dimension $d \ge 0$ and end points $x$ and $a+bT-j$ stays below the linear barrier $a + bt$ for all $t \in [0,T]$. We identify the leading order term as well as the asymptotic error for this probability as $T\to \infty$, depending on $a,b,j,x$. We also derive the behaviour of such leading term as we allow $a,j\to \infty$, and obtain precise bounds for all error terms. Finally, we establish a complementary result where the linear barrier is perturbed by a small concave function.

math.PR↗

arXiv2023

Fermionic Gaussian free field structure in the Abelian sandpile model and uniform spanning tree

In this paper we rigorously construct a finite volume representation for the height-one field of the Abelian sandpile model and the degree field of the uniform spanning tree in terms of the fermionic Gaussian free field. This representation can be seen as the lattice representation of a free symplectic fermion field. It allows us to compute cumulants of those fields, both in finite volume and in the scaling limit, including determining the explicit normalizing constants for fields in the corresponding logarithmic field theory. Furthermore, our results point towards universality of the height-one and degree fields, as we prove that the scaling limits of the cumulants agree (up to constants) in the square and triangular lattice. We also recover the equivalent scaling limits for the hypercubic lattice in higher dimensions, and discuss how to adapt the proofs of our results to general graphs.

math.PR↗

arXiv2022

Absorbing-state phase transition and activated random walks with unbounded capacities

In this article, we study the existence of an absorbing-state phase transition of an Abelian process that generalises the Activated Random Walk (ARW). Given a vertex transitive $G=(V,E)$, we associate to each site $x \in V$ a capacity $w_x \ge 0$, which describes how many inactive particles $x$ can hold, where $\{w_x\}_{x \in V}$ is a collection of i.i.d random variables. When $G$ is an amenable graph, we prove that if $\mathbb E[w_x]<\infty$, the model goes through an absorbing state phase transition and if $\mathbb E[w_x]=\infty$, the model fixates for all $λ>0$. Moreover, in the former case, we provide bounds for the critical density that match the ones available in the classical Activated Random Walk.

math.PR↗

arXiv2022

Stochastic homogenization of Gaussian fields on random media

In this article, we study stochastic homogenization of non-homogeneous Gaussian free fields $\Xi^{g,{\bf a}} $ and bi-Laplacian fields $\Xi^{b,{\bf a}}$. They can be characterized as follows: for $f=\delta$ the solution $u$ of $\nabla \cdot \mathbf{a} \nabla u =f$, ${\bf a}$ is a uniformly elliptic random environment, is the covariance of $\Xi^{g,{\bf a}}$. When $f$ is the white noise, the field $\Xi^{b,{\bf a}}$ can be viewed as the distributional solution of the same elliptic equation. Our results characterize the scaling limit of such fields on both, a sufficiently regular domain $D\subset \mathbb{R}^d$, or on the discrete torus. Based on stochastic homogenization techniques applied to the eigenfunction basis of the Laplace operator $\Delta$, we will show that such families of fields converge to an appropriate multiple of the GFF resp. bi-Laplacian. The limiting fields are determined by their respective homogenized operator $\ahom \Delta$, with constant $\ahom$ depending on the law of the environment ${\bf a}$. The proofs are based on the results found in \cite{Armstrong2019} and \cite{gloria2014optimal}.

math.PR↗

arXiv2021

Constructing fractional Gaussian fields from long-range divisible sandpiles on the torus

In \cite{Cipriani2016}, the authors proved that, with the appropriate rescaling, the odometer of the (nearest neighbours) divisible sandpile on the unit torus converges to a bi-Laplacian field. Here, we study $α$-long-range divisible sandpiles, similar to those introduced in \cite{Frometa2018}. We show that, for $α\in (0,2)$, the limiting field is a fractional Gaussian field on the torus with parameter $α/2$. However, for $α\in [2,\infty)$, we recover the bi-Laplacian field. This provides an alternative construction of fractional Gaussian fields such as the Gaussian Free Field or membrane model using a diffusion based on the generator of Lévy walks. The central tool for obtaining our results is a careful study of the spectrum of the fractional Laplacian on the discrete torus. More specifically, we need the rate of divergence of the eigenvalues as we let the side length of the discrete torus go to infinity. As a side result, we obtain precise asymptotics for the eigenvalues of discrete fractional Laplacians. Furthermore, we determine the order of the expected maximum of the discrete fractional Gaussian field with parameter $γ=\min \{α,2\}$ and $α\in \mathbb{R}_+\backslash\{2\}$ on a finite grid.

math-ph↗

arXiv2021

Fractional Edgeworth expansions for one-dimensional heavy-tailed random variables and applications

In this article, we study a class of lattice random variables in the domain of attraction of an $\alpha$-stable random variable with index $\alpha \in (0,2)$ which satisfy a truncated fractional Edgeworth expansion. Our results include studying the class of such fractional Edgeworth expansions under simple operations, providing concrete examples; sharp rates of convergence to an $\alpha$-stable distribution in a local central limit theorem; Green's function expansions; and finally fluctuations of a class of discrete stochastic PDE's driven by the heavy-tailed random walks belonging to the class of fractional Edgeworth expansions.

math.PR↗