Low-temperature metastability studies

The thesis is divided into four parts as follows. The first two parts are devoted to the analysis of the exit from metastability for a lattice gas at very low temperature and low density that evolves according to the conservative Kawasaki dynamics in a discrete domain $Lambda_beta$ whose volume is exponentially large in the inverse temperature $beta$. Particles perform simple exclusion on $Lambda_beta$ and each of them has a positive activation energy $Delta$, but when they occupy neighbouring sites they feel a binding energy. In the first part of the thesis, we consider variations of the local version of this model, i.e., the gas evolves inside a finite domain $LambdasubsetLambda_beta$ and we investigate how the transition from metastability to stability takes place. We start by considering $L\subsetZ^2$, so that neighbouring particles feel a binding energy $-U_1<0$ in the horizontal direction and $-U_2<0$ in the vertical direction. It turns out that the dynamical behaviour drastically changes whether $U_1=U_2$ (isotropy), $U_1<2U_2$ (weak anisotropy) or $U_1>2U_2$ (strong anisotropy). For all these regimes, our focus is the identification of the critical configurations that have to be crossed with high probability. The derivation of some geometrical properties of the saddles allows us to identify the full geometry of the minimal gates for the nucleation. We observe very different behaviour in the three regimes. Next, we consider the domain $\L$ as a subset of the two-dimensional hexagonal lattice and we assume isotropic interactions between neighbouring particles. We derive the asymptotic behaviour of the transition time from metastability to stability in the limit as $beta$ goes to infinity. We also provide a characterization of the shape of the critical droplets and we emphasize that their description differs from that appearing in the standard square lattice since this feature strongly depends on the underlying geometry. In the second part of the thesis, we deal with the original Kawasaki dynamics on the square lattice in the isotropic regime. This analysis is much harder than in small volumes, indeed now particles are conserved in all the domain and a detailed control of the interaction between droplets and the gas of ``isolated particles'' is needed: the role of the entropy turns out to be crucial. We analyze how subcritical droplets form and dissolve when the volume is ``moderately large'': the evolution of the gas consists of droplets wandering around on multiple space-time scales. Based on these results, we are able to predict that the exit from metastability in ``very large'' volumes occurs via homogeneous nucleation, i.e., a critical droplet appears in a box of moderate volume. In the third part of the thesis, we consider the exit from metastability for systems that evolve under the non-conservative Glauber dynamics, which, contrary to Kawasaki dynamics, has its own peculiar features. In particular, we investigate opinion dynamics on networks with a community structure, assuming that individuals can update their binary opinion as the result of the interactions with an external influence and with other individuals in the network. In the very low temperature regime homogeneous opinion patterns prevail and, as such, it takes evereyone a long time to change opinion. We provide estimates for such a transition time and we fully identify the critical configurations for the dynamics. In the final part, we consider a growing random graph, known as preferential attachment model, such that at each step a new vertex is added and forms $m$ connections. It is well known that the proportion of nodes with a given degree at step $n$ converges to a constant as $n\toinfty$. Our goal is to find the asymptotic distribution of the fluctuations around this limiting value. In particular, we prove a central limit theorem for the joint distribution of all degree counts.