Diffusive scaling of the Kob-Andersen model in Zd\mathbb{Z}^d

We consider the Kob-Andersen model, a cooperative lattice gas with kinetic constraints which has been widely analyzed in the physics literature in connection with the study of the liquid/glass transition. We consider the model in a finite box of linear size $L$ with sources at the boundary. Our result, which holds in any dimension and significantly improves upon previous ones, establishes for any positive vacancy density $q$ a purely diffusive scaling of the relaxation time $T_{\rm rel}$ of the system. Furthermore, as $q\downarrow 0$ we prove upper and lower bounds on $L^{-2} T_{\rm rel} (q,L)$ which agree with the physicists belief that the dominant equilibration mechanism is a cooperative motion of rare large droplets of vacancies. The main tools combine a recent set of ideas and techniques developed to establish universality results for kinetically constrained spin models, with methods from bootstrap percolation, oriented percolation and canonical flows for Markov chains

Kinetically constrained spin models

We analyze the density and size dependence of the relaxation time for kinetically constrained spin models (KCSM) intensively studied in the physical literature as simple models sharing some of the features of a glass transition. KCSM are interacting particle systems on $\Z^d$ with Glauber-like dynamics, reversible w.r.t. a simple product i.i.d Bernoulli($p$) measure. The essential feature of a KCSM is that the creation/destruction of a particle at a given site can occur only if the current configuration of empty sites around it satisfies certain constraints which completely define each specific model. No other interaction is present in the model. From the mathematical point of view, the basic issues concerning positivity of the spectral gap inside the ergodicity region and its scaling with the particle density $p$ remained open for most KCSM (with the notably exception of the East model in $d=1$ \cite{Aldous-Diaconis}). Here for the first time we: i) identify the ergodicity region by establishing a connection with an associated bootstrap percolation model; ii) develop a novel multi-scale approach which proves positivity of the spectral gap in the whole ergodic region; iii) establish, sometimes optimal, bounds on the behavior of the spectral gap near the boundary of the ergodicity region and iv) establish pure exponential decay for the persistence function. Our techniques are flexible enough to allow a variety of constraints and our findings disprove certain conjectures which appeared in the physical literature on the basis of numerical simulations

The East model: recent results and new progresses

The East model is a particular one dimensional interacting particle system in which certain transitions are forbidden according to some constraints depending on the configuration of the system. As such it has received particular attention in the physics literature as a special case of a more general class of systems referred to as kinetically constrained models, which play a key role in explaining some features of the dynamics of glasses. In this paper we give an extensive overview of recent rigorous results concerning the equilibrium and non-equilibrium dynamics of the East model together with some new improvements

Measuring and modelling Internet diffusion using second level domains: the case of Italy

The last 10 years witnessed an exponential growth of the Internet. According to Hobbes' Internet Timeline, the Internet hosts are about 93 million, while in 1989 they were 100,000. The same happens for second level domain names. In July 1989 the registered domains were about 3,900 while they were over 2 million in July 2000. This paper reports about the construction of a database containing daily observations on registrations of second level domain names underneath the it ccTLD in order to analyse the diffusion of Internet among families and businesses. The section of the database referring to domains registered by individuals is analysed. The penetration rate over the relevant population of potential adopters is computed at highly disaggregated geographical level (province). A concentration analysis is carried out to investigate whether the geographical distribution of Internet is less concentrated than population and income suggesting a diffusive effect. Regression analysis is carried out using demographic, social, economic and infrastructure indicators. Finally we briefly describe the further developments of our research. At the present we are constructing a database containing domains registered by firms together with data about the registrants; the idea is to use this new database and the previous one in order to check for the existence of power laws both in the number of domains registered in each province and in the number of domains registered by each firm.Domain names, Internet metrics, Diffusion, Power laws, Zipf s law

Coalescing and branching simple symmetric exclusion process

Motivated by kinetically constrained interacting particle systems (KCM), we consider a reversible coalescing and branching simple exclusion process on a general finite graph $G=(V,E)$ dual to the biased voter model on $G$. Our main goal are tight bounds on its logarithmic Sobolev constant and relaxation time, with particular focus on the delicate slightly supercritical regime in which the equilibrium density of particles tends to zero as $|V|\rightarrow \infty$. Our results allow us to recover very directly and improve to $\ell^p$-mixing, $p\ge 2$, and to more general graphs, the mixing time results of Pillai and Smith for the Fredrickson-Andersen one spin facilitated (FA-$1$f) KCM on the discrete $d$-dimensional torus. In view of applications to the more complex FA-$j$f KCM, $j>1$, we also extend part of the analysis to an analogous process with a more general product state space.Comment: 19 pages, minor change

Universality for critical KCM: finite number of stable directions

In this paper we consider kinetically constrained models (KCM) on $\mathbb Z^2$ with general update families $\mathcal U$. For $\mathcal U$ belonging to the so-called "critical class" our focus is on the divergence of the infection time of the origin for the equilibrium process as the density of the facilitating sites vanishes. In a recent paper Mar\^ech\'e and two of the present authors proved that if $\mathcal U$ has an infinite number of "stable directions", then on a doubly logarithmic scale the above divergence is twice the one in the corresponding $\mathcal U$-bootstrap percolation. Here we prove instead that, contrary to previous conjectures, in the complementary case the two divergences are the same. In particular, we establish the full universality partition for critical $\mathcal U$. The main novel contribution is the identification of the leading mechanism governing the motion of infected critical droplets. It consists of a peculiar hierarchical combination of mesoscopic East-like motions.Comment: 36 pages, 11 figures, expanded and revised presentation, added roadmaps for sections 4 and 5, added and improved figure

Universality results for kinetically constrained spin models in two dimensions

Kinetically constrained models (KCM) are reversible interacting particle systems on $\mathbb Z^d$ with continuous time Markov dynamics of Glauber type, which represent a natural stochastic (and non-monotone) counterpart of the family of cellular automata known as $\mathcal U$-bootstrap percolation. KCM also display some of the peculiar features of the so-called "glassy dynamics", and as such they are extensively used in the physics literature to model the liquid-glass transition, a major and longstanding open problem in condensed matter physics. We consider two-dimensional KCM with update rule $\mathcal U$, and focus on proving universality results for the mean infection time of the origin, in the same spirit as those recently established in the setting of $\mathcal U$-bootstrap percolation. We first identify what we believe are the correct universality classes, which turn out to be different from those of $\mathcal U$-bootstrap percolation. We then prove universal upper bounds on the mean infection time within each class, which we conjecture to be sharp up to logarithmic corrections. In certain cases, including all supercritical models, and the well-known Duarte model, our conjecture has recently been confirmed in [MMT]. In fact, in these cases our upper bound is sharp up to a constant factor in the exponent. For certain classes of update rules, it turns out that the infection time of the KCM diverges much faster than for the corresponding $\mathcal U$-bootstrap process when the equilibrium density of infected sites goes to zero. This is due to the occurrence of energy barriers which determine the dominant behaviour for KCM, but which do not matter for the monotone bootstrap dynamics.Comment: 55 page

Field-Assisted and Thermionic Contributions to Conductance in SnO2 Thick-Films

A deep analysis of conductance in nanostructured SnO2 thick films has been performed. A model for field-assisted thermionic barrier crossing is being proposed to explain the film conductivity. Themodel has been applied to explain the behavior of resistance in vacuum of two sets of nanostructured thick-films with grains having two well-distinct characteristic radii (R = 25nm and R = 125 nm). In the first case the grain radius is shorter than the depletion region width, a limit at which overlapping of barriers takes place, and in the second case it is longer. The behavior of resistance in the presence of dry air has been explained through the mechanism of barrier modulation through gas chemisorption.Fil: Malagú, C.. Universita Di Ferrara; ItaliaFil: Carotta, M. Cristina. Universita Di Ferrara; ItaliaFil: Martinelli, Giuliano. Universita Di Ferrara; ItaliaFil: Ponce, Miguel Adolfo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata. Instituto de Investigaciones en Ciencia y Tecnología de Materiales. Universidad Nacional de Mar del Plata. Facultad de Ingeniería. Instituto de Investigaciones en Ciencia y Tecnología de Materiales; ArgentinaFil: Castro, Miriam Susana. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata. Instituto de Investigaciones en Ciencia y Tecnología de Materiales. Universidad Nacional de Mar del Plata. Facultad de Ingeniería. Instituto de Investigaciones en Ciencia y Tecnología de Materiales; ArgentinaFil: Aldao, Celso Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata. Instituto de Investigaciones en Ciencia y Tecnología de Materiales. Universidad Nacional de Mar del Plata. Facultad de Ingeniería. Instituto de Investigaciones en Ciencia y Tecnología de Materiales; Argentin