arXiv subjects

Vanessa Jacquier

Publications and source records attributed to Vanessa Jacquier.

15 recordsLinked to original sources

arXiv2026

The Ising Model on a Two-Community Stochastic Block Model

We study the Ising model on a two-community stochastic block model, where $n$ spins are split into two equal groups with inter-community interaction parameter $α_n\in[0,1]$. We provide a complete characterization of the phase diagram and show that, almost surely with respect to the graph realization, the model undergoes a uniqueness/non-uniqueness phase transition of the Gibbs measure. In particular, in the supercritical regime, the law of the magnetization vector of the two communities converges to a mixture of Dirac measures that, depending on whether $α_n\gg 1/n$ or $α_n\lesssim1/n$, is supported on two or four points, with possibly different weights. In the uniqueness region, we further analyze the fluctuations of the magnetization vector in the subcritical regime and we prove a quenched central limit theorem.

math.PR↗

arXiv2026

On minimal shapes and isoperimetric constants in hyperbolic lattices

We fully characterize the set of finite shapes with minimal perimeter on hyperbolic lattices given by regular tilings of the hyperbolic plane whose tiles are regular $p$-gons meeting at vertices of degree $q$, with $1/p+1/q<\frac{1}{2}$. In particular, we prove that the ratio between the perimeter and the area (i.e., the number of vertices) of this set of minimal shapes converges to the isoperimetric constant computed in Häggström-Jonasson-Lyons. In fact, our balls which are constructed via layers and not combinatorial balls, will realize the isoperimetric constant for any fixed number of vertices.

math.CO↗

arXiv2026

First-order asymptotics for the structure of the inhomogeneous random graph

In the inhomogeneous random graph model, each vertex $i\in\{1,\ldots,n\}$ is assigned a weight $W_i\sim\text{Unif}(0,1)$, and an edge between any two vertices $i,j$ is present with probability $k(W_i,W_j)/λ_n\in[0,1]$, where $k$ is a positive, symmetric function and $λ_n$ is a scaling parameter that controls the graph density. When $λ_n=1$ (resp.~$λ_n=O(n)$) the typical resulting graph is dense (resp.~sparse). The goal of this paper is the study of structural properties of \textit{large} inhomogeneous random graphs. We focus our attention on graph functions that grow sufficiently slowly as the graph size increases. Under some additional technical assumptions, we show that the first-order asymptotic behavior of all such properties is the same for the inhomogeneous random graph and for the Erdős-Rényi random graph. Our proof relies on two couplings between the inhomogeneous random graph and appropriately constructed Erdős-Rényi random graphs. We demonstrate our method by obtaining asymptotics for two structural properties of the inhomogeneous random graph which were previously unknown. In the sparse regime, we find the leading-order term for the chromatic number. In the dense regime, we find the asymptotics of the so-called $γ$-quasi-clique number.

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

Emergence of metastability on the hyperbolic lattice: Effects of boundary conditions

We investigate the Ising model on finite subgraphs of the hyperbolic lattice under minus boundary conditions and in the presence of a positive external field $h$. Interpreting the boundary as frozen or cold wall conditions, we show that, for small values of $h$, the system exhibits metastable behaviour. Our result is very surprising, since non-amenable graphs, such as hyperbolic lattices, feature exponentially growing boundaries, which typically destabilize local energy minima. In particular, we identify the unique metastable state and characterize the exit time from it. Furthermore, we establish asymptotic results for the distribution of the first hitting time and provide estimates for the spectral gap. Finally, we analyze the energy landscape and describe the nucleation mechanism for values of $h$ outside the metastable regime.

math.PR↗

arXiv2025

Estimate of the exit time for the Long Range Ising model on random regular graphs

We investigate the metastable behavior of the long-range Ising model on random regular graphs under Glauber dynamics at low-temperature. We estimate the energy barrier and exit time from the metastable state using a nontrivial path-wise approach that explicitly accounts for the spatial decay of the interactions and the structural properties of the graph, such as the Cheeger constant and known estimates of the diameter. Our results generalize those of Dommers \cite{dommers2017metastability} for the short-range case, providing a unified framework for understanding metastability in systems with long-range interactions.

math.PR↗

arXiv2025

Exploring Metastability in Ising models: critical droplets, energy barriers and exit time

This paper provides an overview of the research on the metastable behavior of the Ising model. We analyze the transition times from the set of metastable states to the set of the stable states by identifying the critical configurations that the system crosses with high probability during this transition and by computing the energy barrier that the system must overcome to reach the stable state starting from the metastable one. We describe the dynamical phase transition of the Ising model evolving under Glauber dynamics across various contexts, including different lattices, dimensions and anisotropic variants. The analysis is extended to related models, such as long-range Ising model, Blume-Capel and Potts models, as well as to dynamics like Kawasaki dynamics, providing insights into metastability across different systems.

cond-mat.stat-mech↗

arXiv2024

Particle transport based study of nucleation in a ferromagnetic three-state spin system with conservative dynamics

We pose the problem of metastability for a three--state spin system with conservative dynamics. We consider the Blume--Capel model with the Kawasaki dynamics, we prove that, in a particular region of the parameter plane, the metastable state is the unique homogeneous minus state, and we estimate the exit time. To achieve our goal we have to solve several variational problems in the configuration space which result to be particularly involved, due to complicated structure of the trajectories. They key ingredient is the control of the energy differences between the configurations crossed when a spin is transported from the boundary to an internal site of the lattice through a completely arbitrary mixture of the three--state spin species. To master these mechanisms we have introduced a new approach based on the transport of spins along nearest neighbor connected regions of the lattice with constant spin configuration. This novel approach goes beyond the Blume--Capel model and can be used for the study of more general multi--state spin models.

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 $β$. We investigate how the transition between its two maximum-occupancy configurations takes place in the low-temperature regime $β\to\infty$ 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

Homogeneous and heterogeneous nucleation in the three--state Blume--Capel model

The metastable behavior of the stochastic Blume--Capel model with Glauber dynamics is studied when zero-boundary conditions are considered. The presence of zero-boundary conditions changes drastically the metastability scenarios of the model: \emph{heterogeneous nucleation} will be proven in the region of the parameter space where the chemical potential is larger than the external magnetic field.

math-ph↗

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

Metastability of synchronous and asynchronous dynamics

Metastability is an ubiquitous phenomenon in nature, which interests several fields of natural sciences. Its description in the framework of thermodynamics and statistical mechanics has been a taboo for long time since it is a genuine non--equilibrium phenomenon. Since the publication of the first seminal paper in which the metastable behavior of the mean field Curie--Weiss model was approached by means of stochastic techniques, this topic has been largely studied by the scientific community. Several papers and books have been published in which many different spin models were studied and different approaches were developed. In this review we focus on the comparison between the metastable behavior of synchronous and asynchronous dynamics, namely, stochastic processes in discrete time in which at each time either all the spins or one single spin are updated. In particular we discuss how the two different stochastic implementation of the very same Hamiltonian give rise to different metastable behaviors.

cond-mat.stat-mech↗

arXiv2022

Metastability for the Ising model on the hexagonal lattice

We consider the Ising model on the hexagonal lattice evolving according to Metropolis dynamics. We study its metastable behavior in the limit of vanishing temperature when the system is immersed in a small external magnetic field. We determine the asymptotic properties of the transition time from the metastable to the stable state up to a multiplicative factor and study the mixing time and the spectral gap of the Markov process. We give a geometrical description of the critical configurations and show how not only their size but their shape varies depending on the thermodynamical parameters. Finally we provide some results concerning polyiamonds of maximal area and minimal perimeter.

math.PR↗

arXiv2020

Effect of energy degeneracy on the transition time for a series of metastable states: application to Probabilistic Cellular Automata

We consider the problem of metastability for stochastic reversible dynamics with exponentially small transition probabilities. We generalize previous results in several directions. We give an estimate of the spectral gap of the transition matrix and of the mixing time of the associated dynamics in terms of the maximal stability level. These model-independent results hold in particular for a large class of Probabilistic Cellular Automata (PCA), which we then focus on. We consider the PCA in a finite volume, at small and fixed magnetic field, and in the limit of vanishing temperature. This model is peculiar because of the presence of three metastable states, two of which are degenerate with respect to their energy. We identify rigorously the metastable states by giving explicit upper bounds on the stability level of every other configuration. We rely on these estimates to prove a recurrence property of the dynamics, which is a cornerstone of the pathwise approach to metastability. Further, we also identify the metastable states according to the potential-theoretic approach to metastability, and this allows us to give precise asymptotics for the expected transition time from any such metastable state to the stable state.

math.PR↗