arXiv subjects

Francesca R. Nardi

Publications and source records attributed to Francesca R. Nardi.

At least 19 recordsLinked to original sources

arXiv2023

Wireless random-access networks with bipartite interference graphs

We consider random-access networks where nodes represent servers with a queue and can be either active or inactive. A node deactivates at unit rate, while it activates at a rate that depends on its queue length, provided none of its neighbors is active. We consider arbitrary bipartite graphs in the limit as the initial queue lengths become large and identify the transition time between the two states where one half of the network is active and the other half is inactive. The transition path is decomposed into a succession of transitions on complete bipartite subgraphs. We formulate a randomized greedy algorithm that takes the graph as input and gives as output the set of transition paths the network is most likely to follow. Along each path we determine the mean transition time and its law on the scale of its mean. Depending on the activation rates, we identify three regimes of behavior.

math.PR↗

arXiv2023

Weakly interacting oscillators on dense random graphs

We consider a class of weakly interacting particle systems of mean-field type. The interactions between the particles are encoded in a graph sequence, i.e., two particles are interacting if and only if they are connected in the underlying graph. We establish a Law of Large Numbers for the empirical measure of the system that holds whenever the graph sequence is convergent in the sense of graph limits theory, i.e., to a graphon. The limit is shown to be the solution of a non-linear Fokker-Planck equation weighted by the (possibly random) graphon limit. In contrast with the existing literature, our analysis focuses on unlabeled graphons: no regularity assumptions are made on the graph limit and we are able to include general graph sequences such as exchangeable random graphs. Finally, we identify the sequences of graphs, both random and deterministic, for which the associated empirical measure converges to the mean-field limit, i.e., to the solution of a classical McKean-Vlasov equation.

math.PR↗

arXiv2022

Metastability for the degenerate Potts Model with positive external magnetic field under Glauber dynamics

We consider the ferromagnetic q-state Potts model on a finite grid graph with non-zero external field and periodic boundary conditions. The system evolves according to Glauber-type dynamics described by the Metropolis algorithm, and we focus on the low temperature asymptotic regime. We analyze the case of positive external magnetic field. In this energy landscape there are $1$ stable configuration and $q-1$ metastable states. We study the asymptotic behavior of the first hitting time from any metastable state to the stable configuration as $β\to\infty$ in probability, in expectation, and in distribution. We also identify the exponent of the mixing time and find an upper and a lower bound for the spectral gap. We also geometrically identify the union of all minimal gates and the tube of typical trajectories for the transition from any metastable state to the unique stable configuration.

math.PR↗

arXiv2022

Metastability for the degenerate Potts Model with negative external magnetic field under Glauber dynamics

We consider the ferromagnetic q-state Potts model on a finite grid graph with non-zero external field and periodic boundary conditions. The system evolves according to Glauber-type dynamics described by the Metropolis algorithm, and we focus on the low temperature asymptotic regime. We analyze the case of negative external magnetic field. In this scenario there are $q-1$ stable configurations and a unique metastable state. We describe the asymptotic behavior of the first hitting time from the metastable to the set of the stable states as $β\to\infty$ in probability, in expectation, and in distribution. We also identify the exponent of the mixing time and find an upper and a lower bound for the spectral gap. We identify the union of all minimal gates for the transition from the metastable state both to the set of the stable states and to a fixed stable state. Furthermore, we identify the tube of typical trajectories for these two transitions. The accurate knowledge of the energy landscape allows us to give precise asymptotics for the expected transition time from the unique metastable state to the set of the stable configurations.

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

Phase transitions in random mixtures of elementary cellular automata

We investigate one-dimensional Probabilistic Cellular Automata, called Diploid Elementary Cellular Automata (DECA), obtained as random mixture of two different Elementary Cellular Automata rules. All the cells are updated synchronously and the probability for one cell to be $0$ or $1$ at time $t$ depends only on the value of the same cell and that of its neighbors at time $t-1$. These very simple models show a very rich behavior strongly depending on the choice of the two Elementary Cellular Automata that are randomly mixed together and on the parameter which governs probabilistically the mixture. In particular, we study the existence of phase transition for the whole set of possible DECA obtained by mixing the null rule which associates $0$ to any possible local configuration, with any of the other $255$ elementary rule. We approach the problem analytically via a Mean Field approximation and via the use of a rigorous approach based on the application of the Dobrushin Criterion. The distinguishing trait of our result is the possibility to describe the behavior of the whole set of considered DECA without exploiting the local properties of the individual models. The results that we find are coherent with numerical studies already published in the scientific literature and also with some rigorous results proven for some specific models.

cond-mat.stat-mech↗

arXiv2021

Transition time asymptotics of queue-based activation protocols in random-access networks

We consider networks where each node represents a server with a queue. An active node deactivates at unit rate. An inactive node activates at a rate that depends on its queue length, provided none of its neighbors is active. For complete bipartite networks, in the limit as the queues become large, we compute the average transition time between the two states where one half of the network is active and the other half is inactive. We show that the law of the transition time divided by its mean exhibits a trichotomy, depending on the activation rate functions.

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↗

arXiv2018

Tunneling behavior of Ising and Potts models in the low-temperature regime

We consider the ferromagnetic $q$-state Potts model with zero external field in a finite volume and assume that the stochastic evolution of this system is described by a Glauber-type dynamics parametrized by the inverse temperature $β$. Our analysis concerns the low-temperature regime $β\to \infty$, in which this multi-spin system has $q$ stable equilibria, corresponding to the configurations where all spins are equal. Focusing on grid graphs with various boundary conditions, we study the tunneling phenomena of the $q$-state Potts model. More specifically, we describe the asymptotic behavior of the first hitting times between stable equilibria as $β\to \infty$ in probability, in expectation, and in distribution and obtain tight bounds on the mixing time as side-result. In the special case $q=2$, our results characterize the tunneling behavior of the Ising model on grid graphs.

math.PR↗

arXiv2018

Metastability of hard-core dynamics on bipartite graphs

We study the metastable behaviour of a stochastic system of particles with hard-core interactions in a high-density regime. Particles sit on the vertices of a bipartite graph. New particles appear subject to a neighbourhood exclusion constraint, while existing particles disappear, all according to independent Poisson clocks. We consider the regime in which the appearance rates are much larger than the disappearance rates, and there is a slight imbalance between the appearance rates on the two parts of the graph. Starting from the configuration in which the weak part is covered with particles, the system takes a long time before it reaches the configuration in which the strong part is covered with particles. We obtain a sharp asymptotic estimate for the expected transition time, show that the transition time is asymptotically exponentially distributed, and identify the size and shape of the critical droplet representing the bottleneck for the crossover. For various types of bipartite graphs the computations are made explicit. Proofs rely on potential theory for reversible Markov chains, and on isoperimetric results. In a follow-up paper we will use our results to study the performance of random-access wireless networks.

math.PR↗

arXiv2016

Sum of exit times in a series of two metastable states

We consider the problem of non degenerate in energy metastable states forming a series in the framework of reversible finite state space Markov chains. We assume that starting from the state at higher energy the system necessarily visits the second one before reaching the stable state. In this framework, we give a sharp estimate of the exit time from the metastable state at higher energy and, on the proper exponential time scale, we prove an addition rule. As an application of the theory, we study the Blume-Capel model in the zero chemical potential case.

math-ph↗

arXiv2016

Basic Ideas to Approach Metastability in Probabilistic Cellular Automata

Cellular Automata are discrete--time dynamical systems on a spatially extended discrete space which provide paradigmatic examples of nonlinear phenomena. Their stochastic generalizations, i.e., Probabilistic Cellular Automata, are discrete time Markov chains on lattice with finite single--cell states whose distinguishing feature is the \textit{parallel} character of the updating rule. We review some of the results obtained about the metastable behavior of Probabilistic Cellular Automata and we try to point out difficulties and peculiarities with respect to standard Statistical Mechanics Lattice models.

cond-mat.stat-mech↗

arXiv2016

Sum of exit times in series of metastable states in Probabilistic Cellular Automata

Reversible Probabilistic Cellular Automata are a special class of automata whose stationary behavior is described by Gibbs-like measures. For those models the dynamics can be trapped for a very long time in states which are very different from the ones typical of stationarity. This phenomenon can be recasted in the framework of metastability theory which is typical of Statistical Mechanics. In this paper we consider a model presenting two not degenerate in energy metastable states which form a series, in the sense that, when the dynamics is started at one of them, before reaching stationarity, the system must necessarily visit the second one. We discuss a rule for combining the exit times from each of the metastable states.

math-ph↗

arXiv2015

Hitting times asymptotics for hard-core interactions on grids

We consider the hard-core model with Metropolis transition probabilities on finite grid graphs and investigate the asymptotic behavior of the first hitting time between its two maximum-occupancy configurations in the low-temperature regime. In particular, we show how the order-of-magnitude of this first hitting time depends on the grid sizes and on the boundary conditions by means of a novel combinatorial method. Our analysis also proves the asymptotic exponentiality of the scaled hitting time and yields the mixing time of the process in the low-temperature limit as side-result. In order to derive these results, we extended the model-independent framework in [27] for first hitting times to allow for a more general initial state and target subset.

math.PR↗

arXiv2014

A comparison between different cycle decompositions for Metropolis dynamics

In the last decades the problem of metastability has been attacked on rigorous grounds via many different approaches and techniques which are briefly reviewed in this paper. It is then useful to understand connections between different point of views. In view of this we consider irreducible, aperiodic and reversible Markov chains with exponentially small transition probabilities in the framework of Metropolis dynamics. We compare two different cycle decompositions and prove their equivalence.

math.PR↗

arXiv2013

Delay performance in random-access grid networks

We examine the impact of torpid mixing and meta-stability issues on the delay performance in wireless random-access networks. Focusing on regular meshes as prototypical scenarios, we show that the mean delays in an $L\times L$ toric grid with normalized load $ρ$ are of the order $(\frac{1}{1-ρ})^L$. This superlinear delay scaling is to be contrasted with the usual linear growth of the order $\frac{1}{1-ρ}$ in conventional queueing networks. The intuitive explanation for the poor delay characteristics is that (i) high load requires a high activity factor, (ii) a high activity factor implies extremely slow transitions between dominant activity states, and (iii) slow transitions cause starvation and hence excessively long queues and delays. Our proof method combines both renewal and conductance arguments. A critical ingredient in quantifying the long transition times is the derivation of the communication height of the uniformized Markov chain associated with the activity process. We also discuss connections with Glauber dynamics, conductance and mixing times. Our proof framework can be applied to other topologies as well, and is also relevant for the hard-core model in statistical physics and the sampling from independent sets using single-site update Markov chains.

math.PR↗

arXiv2012

Short paths for first passage percolation on the complete graph

We study the complete graph equipped with a topology induced by independent and identically distributed edge weights. The focus of our analysis is on the weight W_n and the number of edges H_n of the minimal weight path between two distinct vertices in the weak disorder regime. We establish novel and simple first and second moment methods using path counting to derive first order asymptotics for the considered quantities. Our results are stated in terms of a sequence of parameters (s_n) that quantifies the extreme-value behaviour of the edge weights, and that describes different universality classes for first passage percolation on the complete graph. These classes contain both n-independent and n-dependent edge weight distributions. The method is most effective for the universality class containing the edge weights E^{s_n}, where E is an exponential(1) random variable and s_n log n -> infty, s_n^2 log n -> 0. We discuss two types of examples from this class in detail. In addition, the class where s_n log n stays finite is studied. This article is a contribution to the program initiated in \cite{BhaHof12}.

math.PR↗