arXiv subjects

Vittoria Silvestri

Publications and source records attributed to Vittoria Silvestri.

14 recordsLinked to original sources

arXiv2026

Gaussian fluctuations for Internal DLA on cylinders

Internal DLA is a discrete random growth model describing growing clusters of particles. Its limiting shape and fluctuations are well understood when the underlying graph is the $d$-dimensional lattice or the cylinder $\mathbb{Z}_N \times \mathbb{Z}$. In the latter geometry, the average fluctuations of IDLA have been shown to converge to the GFF. In this note we generalise this result by showing that, for any vertex-transitive base graph $V_N$ satisfying an eigenvalue convergence condition, the average fluctuations of IDLA on the cylinder $V_N \times \mathbb{Z}$ are given by a GFF. On the way, we present an improved bound on the clusters' maximal fluctuations, which is of independent interest and which implies a shape theorem for IDLA on $V_N \times \mathbb{Z}$ for any vertex-transitive base graph $V_N$.

math.PR↗

arXiv2025

Internal Diffusion Limited Aggregation with Critical Branching Random Walks

Internal Diffusion Limited Aggregation is an interacting particle system that describes the growth of a random cluster governed by the boundary harmonic measure seen from an internal point. Our paper studies IDLA in $\mathbb{Z}^d$ driven by critical branching random walks. We prove that, unlike classical IDLA, this process exhibits a phase transition in the dimension. More precisely, we establish the existence of a spherical shape theorem in dimension $d\geq 3$ and the absence of a spherical shape theorem for $d \leq 2$. Our bounds on the inner and outer worst deviations are of polynomial nature, which we expect to be a feature of this model.

math.PR↗

arXiv2024

A martingale approach to time-dependent and time-periodic linear response in Markov jump processes

We consider a Markov jump process on a general state space to which we apply a time-dependent weak perturbation over a finite time interval. By martingale-based stochastic calculus, under a suitable exponential moment bound for the perturbation we show that the perturbed process does not explode almost surely and we study the linear response (LR) of observables and additive functionals. When the unperturbed process is stationary, the above LR formulas become computable in terms of the steady state two-time correlation function and of the stationary distribution. Applications are discussed for birth and death processes, random walks in a confining potential, random walks in a random conductance field. We then move to a Markov jump process on a finite state space and investigate the LR of observables and additive functionals in the oscillatory steady state (hence, over an infinite time horizon), when the perturbation is time-periodic. As an application we provide a formula for the complex mobility matrix of a random walk on a discrete $d$-dimensional torus, with possibly heterogeneous jump rates.

math.PR↗

arXiv2023

Universality Conjectures for Activated Random Walk

Activated Random Walk is a particle system displaying Self-Organized Criticality, in that the dynamics spontaneously drive the system to a critical state. How universal is this critical state? We state many interlocking conjectures aimed at different aspects of this question: scaling limits, microscopic limits, temporal and spatial mixing, incompressibility, and hyperuniformity.

math.PR↗

arXiv2022

Explosive growth for a constrained Hastings-Levitov aggregation model

We consider a constrained version of the HL$(0)$ Hastings--Levitov model of aggregation in the complex plane, in which particles can only attach to the part of the cluster that has already been grown. Although one might expect that this gives rise to a non-trivial limiting shape, we prove that the cluster grows explosively: in the upper half plane, the aggregate accumulates infinite diameter as soon as it reaches positive capacity. More precisely, we show that after $nt$ particles of (half-plane) capacity $1/(2n)$ have attached, the diameter of the shape is highly concentrated around $\sqrt{t\log n}$, uniformly in $t\in [0,T]$. This illustrates a new instability phenomenon for the growth of single trees/fjords in unconstrained HL$(0)$.

math.PR↗

arXiv2021

Scaling limits for planar aggregation with subcritical fluctuations

We study scaling limits of a family of planar random growth processes in which clusters grow by the successive aggregation of small particles. In these models, clusters are encoded as a composition of conformal maps and the location of each successive particle is distributed according to the density of harmonic measure on the cluster boundary, raised to some power. We show that, when this power lies within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a limit of stability. The methodology developed in this paper provides a blueprint for analysing more general random growth models, such as the Hastings-Levitov family.

math.PR↗

arXiv2021

How far do Activated Random Walkers spread from a single source?

Unlike many particle systems, Activated Random Walk has nontrivial behavior even in one spatial dimension. We prove inner and outer bounds on the spread of n activated random walkers from a single source in Z. The inner bound involves a comparison with the stationary distribution of activated random walkers on a finite interval, while the outer bound involves a comparison with the stabilization of an infinite Bernoulli configuration of activated random walkers on Z.

math.PR↗

arXiv2021

Stability of regularized Hastings-Levitov aggregation in the subcritical regime

We prove bulk scaling limits and fluctuation scaling limits for a two-parameter class ALE$(α,η)$ of continuum planar aggregation models. The class includes regularized versions of the Hastings--Levitov family HL$(α)$ and continuum versions of the family of dielectric breakdown models, where the local attachment intensity for new particles is specified as a negative power $-η$ of the density of arc length with respect to harmonic measure. The limit dynamics follow solutions of a certain Loewner--Kufarev equation, where the driving measure is made to depend on the solution and on the parameter $ζ=α+η$. Our results are subject to a subcriticality condition $ζ\le1$: this includes HL$(α)$ for $α\le1$ and also the case $α=2,η=-1$ corresponding to a continuum Eden model. Hastings and Levitov predicted a change in behaviour for HL$(α)$ at $α=1$, consistent with our results. In the regularized regime considered, the fluctuations around the scaling limit are shown to be Gaussian, with independent Ornstein--Uhlenbeck processes driving each Fourier mode, which are seen to be stable if and only if $ζ\le1$.

math.PR↗

arXiv2020

Internal DLA on cylinder graphs: fluctuations and mixing

We use coupling ideas introduced in \cite{levine2018long} to show that an IDLA process on a cylinder graph $G\times \mathbb{Z}$ forgets a typical initial profile in $\mathcal{O}( N\sqrt{τ_N} (\log \! N)^2 )$ steps for large $N$, where $N$ is the size of the base graph $G$, and $τ_N$ is the total variation mixing time of a simple random walk on $G$. The main new ingredient is a maximal fluctuations bound for IDLA on $G\times \mathbb{Z}$ which only relies on the mixing properties of the base graph $G$ and the Abelian property.

math.PR↗

arXiv2018

How long does it take for Internal DLA to forget its initial profile?

Internal DLA is a discrete model of a moving interface. On the cylinder graph $\mathbb{Z}_N \times \mathbb{Z}$, a particle starts uniformly on $\mathbb{Z}_N \times \{0\}$ and performs simple random walk on the cylinder until reaching an unoccupied site in $\mathbb{Z}_N \times \mathbb{Z}_{\geq 0}$, which it occupies forever. This operation defines a Markov chain on subsets of the cylinder. We first show that a typical subset is rectangular with at most logarithmic fluctuations. We use this to prove that two Internal DLA chains started from different typical subsets can be coupled with high probability by adding order $N^2 \log N$ particles. For a lower bound, we show that at least order $N^2$ particles are required to forget which of two independent typical subsets the process started from.

math.PR↗

arXiv2017

Fluctuation theorems for discrete kinetic models of molecular motors

Motivated by discrete kinetic models for non-cooperative molecular motors on periodic tracks, we consider random walks (also not Markov) on quasi one dimensional (1d) lattices, obtained by gluing several copies of a fundamental graph in a linear fashion. We show that, for a suitable class of quasi 1d lattices, the large deviation rate function associated to the position of the walker satisfies a Gallavotti-Cohen symmetry for any choice of the dynamical parameters defining the stochastic walk. This class includes the linear model considered in \cite{LLM1}. We also derive fluctuation theorems for the time-integrated cycle currents and discuss how the matrix approach of \cite{LLM1} can be extended to derive the above Gallavotti-Cohen symmetry for any Markov random walk on $\mathbb{Z}$ with periodic jump rates. Finally, we review in the present context some large deviation results of \cite{FS1} and give some specific examples with explicit computations.

cond-mat.stat-mech↗

arXiv2015

Fluctuation results for Hastings-Levitov planar growth

We study the fluctuations of the outer domain of Hastings-Levitov clusters in the small particle limit. These are shown to be given by a continuous Gaussian process $\mathcal{F}$ taking values in the space of holomorphic functions on $\{ |z|>1 \}$, of which we provide an explicit construction. The boundary values $\mathcal{W}$ of $\mathcal{F}$ are shown to perform an Ornstein-Uhlenbeck process on the space of distributions on the unit circle $\mathbb{T}$, which can be described as the solution to the stochastic fractional heat equation \[ \frac{\partial}{\partial t} \mathcal{W} (t,\vartheta ) = - (-Δ)^{1/2} \mathcal{W} (t,\vartheta ) + \sqrt{2}\, ξ(t, \vartheta ) \,, \] where $Δ$ denotes the Laplace operator acting on the spatial component, and $ξ(t,\vartheta )$ is a space-time white noise. As a consequence we find that, when the cluster is left to grow indefinitely, the boundary process $\mathcal{W}$ converges to a log-correlated Fractional Gaussian Field, which can be realised as $(-Δ)^{-1/4}W$, for $W$ complex White Noise on $\mathbb{T}$.

math.PR↗

arXiv2014

Discrete kinetic models for molecular motors: asymptotic velocity and gaussian fluctuations

We consider random walks on quasi one dimensional lattices, as introduced in \cite{FS}. This mathematical setting covers a large class of discrete kinetic models for non-cooperative molecular motors on periodic tracks. We derive general formulas for the asymptotic velocity and diffusion coefficient, and we show how to reduce their computation to suitable linear systems of the same degree of a single fundamental cell, with possible linear chain removals. We apply the above results to special families of kinetic models, also catching some errors in the biophysics literature.

math.PR↗

arXiv2014

Random walks on quasi one dimensional lattices: large deviations and fluctuation theorems

Several stochastic processes modeling molecular motors on a linear track are given by random walks (not necessarily Markovian) on quasi 1d lattices and share a common regenerative structure. Analyzing this abstract common structure, we derive information on the large fluctuations of the stochastic process by proving large deviation principles for the first--passage times and for the position. We focus our attention on the Gallavotti-Cohen-type symmetry of the position rate function (fluctuation theorem), showing its equivalence with the independence of suitable random variables. In the special case of Markov random walks, we show that this symmetry is universal only inside a suitable class of quasi 1d lattices.

math.PR↗