arXiv subjects

Francesco Manzo

Publications and source records attributed to Francesco Manzo.

7 recordsLinked to original sources

arXiv2024

Quasi-stationary distributions of non-absorbing Markov chains

We consider reversible ergodic Markov chains with finite state space, and we introduce a new notion of quasi-stationary distribution that does not require the presence of any absorbing state. In our setting, the hitting time of the absorbing set is replaced by an optimal strong stationary time, representing the ``hitting time of the stationary distribution''. On the one hand, we show that our notion of quasi-stationary distribution corresponds to the natural generalization of the \emph{Yaglom limit}. On the other hand, similarly to the classical quasi-stationary distribution, we show that it can be written in terms of the eigenvectors of the underlying Markov kernel, and it is therefore amenable of a geometric interpretation. Moreover, we recover the usual exponential behavior that characterizes quasi-stationary distributions and metastable systems. We also provide some toy examples, which show that the phenomenology is richer compared to the absorbing case. Finally, we present some counterexamples, showing that the assumption on the reversibility cannot be weakened in general.

math.PR↗

arXiv2021

A probabilistic proof of Cooper and Frieze's "First Visit Time Lemma"

In this short note we present an alternative proof of the so-called First Visit Time Lemma (FVTL), originally presented by Cooper and Frieze in its first formulation in [21], and then used and refined in a list of papers by Cooper, Frieze and coauthors. We work in the original setting, considering a growing sequence of irreducible Markov chains on $n$ states. We assume that the chain is rapidly mixing and with a stationary measure having no entry which is too small nor too large. Under these assumptions, the FVTL shows the exponential decay of the distribution of the hitting time of a given state $x$ -- for the chain started at stationarity -- up to a small multiplicative correction. While the proof of the FVTL presented by Cooper and Frieze is based on tools from complex analysis, and it requires an additional assumption on a generating function, we present a completely probabilistic proof, relying on the theory of quasi-stationary distributions and on strong-stationary times arguments. In addition, under the same set of assumptions, we provide some quantitative control on the Doob's transform of the chain on the complement of the state $x$.

math.PR↗

arXiv2016

Strong times and first hitting

We generalize the notion of strong stationary time and we give a representation formula for the hitting time to a target set in the general case of non-reversible Markov processes.

math.PR↗

arXiv2015

Conditioned, quasi-stationary, restricted measures and escape from metastable states

We study the asymptotic hitting time $τ^{(n)}$ of a family of Markov processes $X^{(n)}$ to a target set $G^{(n)}$ when the process starts from a trap defined by very general properties. We give an explicit description of the law of $X^{(n)}$ conditioned to stay within the trap, and from this we deduce the exponential distribution of $τ^{(n)}$. Our approach is very broad ---it does not require reversibility, the target $G$ does not need to be a rare event, and the traps and the limit on $n$ can be of very general nature--- and leads to explicit bounds on the deviations of $τ^{(n)}$ from exponentially. We provide two non trivial examples to which our techniques directly apply.

math.PR↗

arXiv2013

Nucleation and growth for the Ising model in $d$ dimensions at very low temperatures

This work extends to dimension $d\geq3$ the main result of Dehghanpour and Schonmann. We consider the stochastic Ising model on ${\mathbb{Z}}^d$ evolving with the Metropolis dynamics under a fixed small positive magnetic field $h$ starting from the minus phase. When the inverse temperature $β$ goes to $\infty$, the relaxation time of the system, defined as the time when the plus phase has invaded the origin, behaves like $\exp({βκ_d})$. The value $κ_d$ is equal to \[κ_d=\frac{1}{d+1}(Γ_1+\cdots+Γ_d),\] where $Γ_i$ is the energy of the $i$-dimensional critical droplet of the Ising model at zero temperature and magnetic field $h$.

math.PR↗

arXiv2010

A d dimensional nucleation and growth model

We analyze the relaxation time of a ferromagnetic d dimensional growth model on the lattice. The model is characterized by d param- eters which represent the activation energies of a site, depending on the number of occupied nearest neighbours. This model is a natural generalisation of the model studied by Dehghanpour and Schonmann [DS97a], where the activation energy of a site with more than two occupied neighbours is zero.

math.PR↗

arXiv2001

Metastability in Glauber dynamics in the low-temperature limit: Beyond exponential asymptotics

We consider Glauber dynamics of classical spin systems of Ising type in the limit when the temperature tends to zero in finite volume. We show that information on the structure of the most profound minima and the connecting saddle points of the Hamiltonian can be translated into sharp estimates on the distribution of the times of metastable transitions between such minima as well as the low lying spectrum of the generator. In contrast with earlier results on such problems, where only the asymptotics of the exponential rates is obtained, we compute the precise pre-factors up to multiplicative errors that tend to 1 as $T\downarrow 0$. As an example we treat the nearest neighbor Ising model on the 2 and 3 dimensional square lattice. Our results improve considerably earlier estimates obtained by Neves-Schonmann and Ben Arous-Cerf. Our results employ the methods introduced by Bovier, Eckhoff, Gayrard, and Klein.

cond-mat.dis-nn↗