arXiv subjects

Simone Baldassarri

Publications and source records attributed to Simone Baldassarri.

17 recordsLinked to original sources

arXiv2026

Parameter inference for partially observed branching processes

In this paper, we study an age-dependent branching process. In the simplest setting, the population is divided into two age groups, namely juveniles and adults. Our objective is to estimate the model parameters using observations of the total population size only (i.e., juveniles plus adults). Focusing on the ergodic regime of the model, we introduce a method-of-moments estimator and establish its asymptotic normality. Several extensions are discussed, including models with more than two age groups.

math.ST↗

arXiv2026

Optimal strategies for controlled growth in metastable Kawasaki dynamics

In this paper, we develop a Markov decision process (MDP) formulation for the low--temperature metastable Ising model evolving according to Kawasaki dynamics in a finite box of the two--dimensional square lattice. We analyze how an external controller can guide the system to the all--occupied state by appropriately adding and moving particles at specified moments in time. To this end, we construct a reduced MDP on a constrained family of configurations having a single cluster, a regime where particle attachment is more likely than detachment. We investigate two reward structures: one that depends solely on the time to reach the target configuration, and another that incorporates action--dependent energy costs. Within this MDP framework, we characterize the exact optimal policies under both reward structures, which turn out to have a different behavior: while a purely efficiency--based criterion promotes the growth from the boundary centers of the cluster, an energy--based reward function favours the growth at the corners of the cluster.

math.OC↗

arXiv2026

Infection models on dense dynamic random graphs

We consider Susceptible-Infected-Recovered (SIR) models on dense dynamic random graphs, in which the joint dynamics of vertices and edges are co-evolutionary, i.e., they influence each other bidirectionally. In particular, edges appear and disappear over time depending on the states of the two connected vertices, on how long they have been infected, and on the total density of susceptible and infected vertices. Our main results establish functional laws of large numbers for the densities of susceptible, infected, and recovered vertices, jointly with the underlying evolving random graphs in the graphon space. Our results are supported by simulations, which characterize the limiting size of the epidemics, i.e., the limiting density of susceptible vertices, and how the peak of the epidemics depends on the rate of the evolution of the underlying graph. The proofs of our main results rely on the careful construction of a mimicking process, obtained by approximating the two-way feedback interaction between vertex and edge dynamics with a mean-field type interaction, acting only as one-way feedback, that remains sufficiently close to the original co-evolutionary process. To treat the more general setting in which edge dynamics are affected by the proportions of susceptible and infected individuals, we introduce a methodological extension of existing techniques. We thus show that our model exhibits multiple epidemic peaks -- a phenomenon observed in real-world epidemics -- which can emerge in models that incorporate mutual feedback between vertex and edge dynamics.

math.PR↗

arXiv2026

Metastable opinion dynamics with hidden preferences: an Ising model with neutral agents

We introduce a new Ising-type framework for opinion dynamics that explicitly separates private preferences from publicly expressed binary opinions and naturally incorporates neutral agents. Each individual is endowed with an immutable hidden preference, while public opinions evolve through Metropolis dynamics on a finite graph. This formulation extends classical sociophysical Ising models by capturing the tension between internal conviction, social conformity, and neutrality. Focusing on highly symmetric grid networks and spatially structured hidden-preference patterns, we analyze the resulting low-temperature dynamics using the pathwise approach to metastability. We provide a complete characterization of stable and metastable configurations, identify the maximal stability level of the energy landscape, and derive sharp asymptotics for hitting and mixing times. A central technical contribution is a new family of isoperimetric inequalities for polyominoes on the torus, which emerge from a geometric representation of opinion clusters and play a key role in determining critical configurations and energy barriers. Our results provide a quantitative understanding of how spatial heterogeneity in hidden preferences qualitatively reshapes collective opinion transitions and illustrate the power of geometric and probabilistic methods in the study of complex interacting systems.

math.PR↗

arXiv2025

Parameter estimation in interacting particle systems on dynamic random networks

In this paper we consider a class of interacting particle systems on dynamic random networks, in which the joint dynamics of vertices and edges acts as one-way feedback, i.e., edges appear and disappear over time depending on the state of the two connected vertices, while the vertex dynamics does not depend on the edge process. Our goal is to estimate the underlying dynamics from partial information of the process, specifically from snapshots of the total number of edges present. We showcase the effectiveness of our inference method through various numerical results.

math.PR↗

arXiv2025

Functional central limit theorem for the subgraph count of the voter model on dynamic random graphs

In this paper we consider two-opinion voter models on dynamic random graphs, in which the joint dynamics of opinions and graphs acts as one-way feedback, i.e., edges appear and disappear over time depending on the opinions of the two connected vertices, while the opinion dynamics is not affected by the graph structure. Our goal is to investigate the joint evolution of the entries of a voter subgraph count vector, i.e., vector of subgraphs where each vertex has a specific opinion, in the regime that the number of vertices grows large. The main result of this paper is a functional central limit theorem. In particular, we prove that, under a proper centering and scaling, the joint functional of the vector of subgraph counts converges to a specific multidimensional Gaussian process.

math.PR↗

arXiv2024

Grassmannian calculus for probability

The present overview and gentle introduction to Grassmannian calculus and some of its applications to probability collects the notes of a mini-course given by the authors at the Brazilian School of Probability, August 5-9, 2024, in Salvador, Bahia, Brazil. The content is by no means comprehensive, and is a personal summary and interpretation of results and applications of this interesting area of research.

math.PR↗

arXiv2024

Opinion dynamics on dense dynamic random graphs

We consider two-opinion voter models on dense dynamic random graphs. Our goal is to understand and describe the occurrence of consensus versus polarisation over long periods of time. The former means that all vertices have the same opinion, the latter means that the vertices split into two communities with different opinions and few disagreeing edges. We consider three models for the joint dynamics of opinions and graphs: one with a one-way feedback and two which are co-evolutionary, i.e., with a two-way feedback. In the first model only coexistence is attainable, meaning that both opinions survive, but with the presence of many disagreeing edges. In the second model only consensus prevails, while in the third model polarisation is possible. Our main results are functional laws of large numbers for the densities of the two opinions, functional laws of large numbers for the dynamic random graphs in the space of graphons, and a characterisation of the limiting densities in terms of Beta-distributions. Our results are supported by simulations. To prove our results we develop a novel method that involves coupling the co-evolutionary process to a mimicking process with one-way feedback. We expect that this method can be extended to other dense co-evolutionary models.

math.PR↗

arXiv2024

Droplet dynamics in a two-dimensional rarefied gas under Kawasaki dynamics

This is the second in a series of three papers in which we study a lattice gas subject to Kawasaki conservative dynamics at inverse temperature $β>0$ in a large finite box $Λ_β\subset\mathbb Z^2$ whose volume depends on $β$. Each pair of neighbouring particles has a negative binding energy $-U<0$, while each particle has a positive activation energy $Δ>0$. The initial configuration is drawn from the grand-canonical ensemble restricted to the set of configurations where all the droplets are subcritical. Our goal is to describe, in the metastable regime $Δ\in(U,2U)$ and in the limit as $β\to\infty$, how and when the system nucleates. In the first paper we showed that subcritical droplets behave as quasi-random walks. In the present paper we use the results in the first paper to analyse how subcritical droplets form and dissolve on multiple space-time scales when the volume is moderately large, i.e., $|Λ_β|=\mathrm e^{Θβ}$ with $Δ<Θ<2Δ-U$. In the third paper we consider the setting where the volume is very large, namely, $|Λ_β|=\mathrm e^{Θβ}$ with $Δ<Θ<Γ-(2Δ-U)$, where $Γ$ is the energy of the critical droplet in the local model with fixed volume, and use the results in the first two papers to identify the nucleation time. We will see that in a very large volume critical droplets appear more or less independently in boxes of moderate volume, a phenomenon referred to as homogeneous nucleation. Since Kawasaki dynamics is conservative, i.e., particles are preserved, we need to control non-local effects in the way droplets are formed and dissolved. This is done via a deductive approach: the tube of typical trajectories leading to nucleation is described via a series of events on which the evolution of the gas consists of droplets wandering around on multiple space-time scales.

math.PR↗

arXiv2024

Homogeneous nucleation for two-dimensional Kawasaki dynamics

This is the third in a series of three papers in which we study a lattice gas subject to Kawasaki dynamics at inverse temperature $\beta>0$ in a large finite box $\Lambda_\beta \subset \mathbb{Z}^2$ whose volume depends on $\beta$. Each pair of neighbouring particles has a negative binding energy $-U<0$, while each particle has a positive activation energy $\Delta>0$. The initial configuration is drawn from the grand-canonical ensemble restricted to the set of configurations where all the droplets are subcritical. Our goal is to describe, in the metastable regime $\Delta \in (U,2U)$ and in the limit as $\beta\to\infty$, how and when the system nucleates, i.e., creates a critical droplet somewhere in $\Lambda_\beta$ that subsequently grows by absorbing particles from the surrounding gas. In the first paper we showed that subcritical droplets behave as quasi-random walks. In the second paper we used the results in the first paper to analyse how subcritical droplets form and dissolve on multiple space-time scales when the volume is moderately large, namely, $|\Lambda_\beta| = \mathrm{e}^{\theta\beta}$ with $\Delta < \theta < 2\Delta-U$. In the present paper we consider the setting where the volume is very large, namely, $|\Lambda_\beta| = \mathrm{e}^{\Theta\beta}$ with $\Theta < \Gamma-(2\Delta-U)$, where $\Gamma$ is the energy of the critical droplet in the local model with fixed volume, and use the results in the first two papers to identify the nucleation time and the tube of typical trajectories towards nucleation. We will see that in a very large volume critical droplets appear more or less independently in boxes of moderate volume, a phenomenon referred to as homogeneous nucleation. One of the key ingredients in the proof is an estimate showing that no information can travel between these boxes on relevant time scales.

math.PR↗

arXiv2024

Asymptotic normality of degree counts in a general preferential attachment model

We consider the preferential attachment model. This is a growing random graph such that at each step a new vertex is added and forms $m$ connections. The neighbors of the new vertex are chosen at random with probability proportional to their degree. It is well known that the proportion of nodes with a given degree at step $n$ converges to a constant as $n\rightarrow\infty$. The goal of this paper is to investigate the asymptotic distribution of the fluctuations around this limiting value. We prove a central limit theorem for the joint distribution of all degree counts. In particular, we give an explicit expression for the asymptotic covariance. This expression is rather complex, so we compute it numerically for various parameter choices. We also use numerical simulations to argue that the convergence is quite fast. The proof relies on the careful construction of an appropriate martingale.

math.PR↗

arXiv2023

Critical configurations of the hard-core model on square grid graphs

We consider the hard-core model on a finite square grid graph with stochastic Glauber dynamics parametrized by the inverse temperature $\beta$. We investigate how the transition between its two maximum-occupancy configurations takes place in the low-temperature regime $\beta\to\infty$ in the case of periodic boundary conditions. The hard-core constraints and the grid symmetry make the structure of the critical configurations, also known as essential saddles, for this transition very rich and complex. We provide a comprehensive geometrical characterization of the set of critical configurations that are asymptotically visited with probability one. In particular, we develop a novel isoperimetric inequality for hard-core configurations with a fixed number of particles and we show how not only their size but also their shape determines the characterization of the saddles.

math.PR↗

arXiv2023

Metastability for Kawasaki dynamics on the hexagonal lattice

In this paper we analyze the metastable behavior for the Ising model that evolves under Kawasaki dynamics on the hexagonal lattice $\mathbb{H}^2$ in the limit of vanishing temperature. Let $Λ\subset\mathbb{H}^2$ a finite set which we assume to be arbitrarily large. Particles perform simple exclusion on $Λ$, but when they occupy neighboring sites they feel a binding energy $-U<0$. Along each bond touching the boundary of $Λ$ from the outside to the inside, particles are created with rate $ρ=e^{-Δβ}$, while along each bond from the inside to the outside, particles are annihilated with rate 1, where $β$ is the inverse temperature and $Δ>0$ is an activity parameter. For the choice $Δ\in{(U,\frac{3}{2}U)}$ we prove that the empty (resp.\ full) hexagon is the unique metastable (resp.\ stable) state. We determine the asymptotic properties of the transition time from the metastable to the stable state and we give a description of the critical configurations. We show how not only their size but also their shape varies depending on the thermodynamical parameters. Moreover, we emphasize the role that the specific lattice plays in the analysis of the metastable Kawasaki dynamics by comparing the different behavior of this system with the corresponding system on the square lattice.

math.PR↗

arXiv2022

Ising model on clustered networks: A model for opinion dynamics

We study opinion dynamics on networks with a nontrivial community structure, assuming individuals can update their binary opinion as the result of the interactions with an external influence with strength $h\in [0,1]$ and with other individuals in the network. To model such dynamics, we consider the Ising model with an external magnetic field on a family of finite networks with a clustered structure. Assuming a unit strength for the interactions inside each community, we assume that the strength of interaction across different communities is described by a scalar $ε\in [-1,1]$, which allows a weaker but possibly antagonistic effect between communities. We are interested in the stochastic evolution of this system described by a Glauber-type dynamics parameterized by the inverse temperature $β$. We focus on the low-temperature regime $β\rightarrow\infty$, in which homogeneous opinion patterns prevail and, as such, it takes the network a long time to fully change opinion. We investigate the different metastable and stable states of this opinion dynamics model and how they depend on the values of the parameters $ε$ and $h$. More precisely, using tools from statistical physics, we derive rigorous estimates in probability, expectation, and law for the first hitting time between metastable (or stable) states and (other) stable states, together with tight bounds on the mixing time and spectral gap of the Markov chain describing the network dynamics. Lastly, we provide a full characterization of the critical configurations for the dynamics, i.e., those which are visited with high probability along the transitions of interest.

math.PR↗

arXiv2022

Critical Droplets and sharp asymptotics for Kawasaki dynamics with strongly anisotropic interactions

In this paper we analyze metastability and nucleation in the context of the Kawasaki dynamics for the two-dimensional Ising lattice gas at very low temperature. Let $Λ\subset\mathbb{Z}^2$ be a finite box. Particles perform simple exclusion on $Λ$, but when they occupy neighboring sites they feel a binding energy $-U_1<0$ in the horizontal direction and $-U_2<0$ in the vertical one. Thus the Kawasaki dynamics is conservative inside the volume $Λ$. Along each bond touching the boundary of $Λ$ from the outside to the inside, particles are created with rate $ρ=e^{-Δβ}$, while along each bond from the inside to the outside, particles are annihilated with rate $1$, where $β>0$ is the inverse temperature and $Δ>0$ is an activity parameter. Thus, the boundary of $Λ$ plays the role of an infinite gas reservoir with density $ρ$. We consider the parameter regime $U_1>2U_2$ also known as the strongly anisotropic regime. We take $Δ\in{(U_1,U_1+U_2)}$, so that the empty (respectively full) configuration is a metastable (respectively stable) configuration. We investigate how the transition from empty to full takes place with particular attention to the critical configurations that asymptotically have to be crossed with probability 1. The derivation of some geometrical properties of the saddles allows us to identify the full geometry of the minimal gates and their boundaries for the nucleation in the strongly anisotropic case. We observe very different behaviors for this case with respect to the isotropic ($U_1=U_2$) and weakly anisotropic ($U_1<2U_2$) ones. Moreover, we derive mixing time, spectral gap and sharp estimates for the asymptotic transition time for the strongly anisotropic case.

math.PR↗

arXiv2022

Critical Droplets and sharp asymptotics for Kawasaki dynamics with weakly anisotropic interactions. Extended version

In this paper we analyze metastability and nucleation in the context of the Kawasaki dynamics for the two-dimensional Ising lattice gas at very low temperature with periodic boundary conditions. Let $β>0$ be the inverse temperature and let $Λ\subsetΛ^β\subset\mathbb{Z}^2$ be two boxes. We consider the asymptotic regime corresponding to the limit as $β\rightarrow\infty$ for finite volume $Λ$ and $\lim_{β\rightarrow\infty}\frac{1}β\log|Λ^β|=\infty$. We study the simplified model, in which particles perform independent random walks on $Λ^β\setminusΛ$ and inside $Λ$ particles perform simple exclusion, but when they occupy neighboring sites they feel a binding energy $-U_1<0$ in the horizontal direction and $-U_2<0$ in the vertical one. Thus the Kawasaki dynamics is conservative inside the volume $Λ^β$. The initial configuration is chosen such that $Λ$ is empty and $ρ|Λ^β|$ particles are distributed randomly over $Λ^β\setminusΛ$. Our results will use a deep analysis of a local model, i.e., particles perform Kawasaki dynamics inside $Λ$ and along each bond touching the boundary of $Λ$ from the outside to the inside, particles are created with rate $ρ=e^{-Δβ}$, while along each bond from the inside to the outside, particles are annihilated with rate $1$, where $Δ>0$ is an activity parameter. Thus, in the local model the boundary of $Λ$ plays the role of an infinite gas reservoir with density $ρ$. We take $Δ\in{(U_1,U_1+U_2)}$, so that the empty (respectively full) configuration is a metastable (respectively stable) configuration. We investigate how the transition from empty to full takes place in the local model with particular attention to the critical configurations that asymptotically have to be crossed with probability 1.

math.PR↗

arXiv2021

Metastability in a lattice gas with strong anisotropic interactions under Kawasaki dynamics

In this paper we analyze metastability and nucleation in the context of a local version of the Kawasaki dynamics for the two-dimensional strongly anisotropic Ising lattice gas at very low temperature. Let $Λ\subset\mathbb{Z}^2$ be a finite box. Particles perform simple exclusion on $Λ$, but when they occupy neighboring sites they feel a binding energy $-U_1<0$ in the horizontal direction and $-U_2<0$ in the vertical one. Thus the Kawasaki dynamics is conservative inside the volume $Λ$. Along each bond touching the boundary of $Λ$ from the outside to the inside, particles are created with rate $ρ=e^{-Δβ}$, while along each bond from the inside to the outside, particles are annihilated with rate $1$, where $β$ is the inverse temperature and $Δ>0$ is an activity parameter. Thus, the boundary of $Λ$ plays the role of an infinite gas reservoir with density $ρ$. We consider the parameter regime $U_1>2U_2$ also known as the strongly anisotropic regime. We take $Δ\in{(U_1,U_1+U_2)}$ and we prove that the empty (respectively full) configuration is a metastable (respectively stable) configuration. We consider the asymptotic regime corresponding to finite volume in the limit of large inverse temperature $β$. We investigate how the transition from empty to full takes place. In particular, we estimate in probability, expectation and distribution the asymptotic transition time from the metastable configuration to the stable configuration. Moreover, we identify the size of the \emph{critical droplets}, as well as some of their properties. We observe very different behavior in the weakly and strongly anisotropic regimes. We find that the \emph{Wulff shape}, i.e., the shape minimizing the energy of a droplet at fixed volume, is not relevant for the nucleation pattern.

math.PR↗