Asymptotically exponential hitting times and metastability: a pathwise approach without reversibility

We study the hitting times of Markov processes to target set $G$, starting from a reference configuration $x_0$ or its basin of attraction. The configuration $x_0$ can correspond to the bottom of a (meta)stable well, while the target $G$ could be either a set of saddle (exit) points of the well, or a set of further (meta)stable configurations. Three types of results are reported: (1) A general theory is developed, based on the path-wise approach to metastability, which has three important attributes. First, it is general in that it does not assume reversibility of the process, does not focus only on hitting times to rare events and does not assume a particular starting measure. Second, it relies only on the natural hypothesis that the mean hitting time to $G$ is asymptotically longer than the mean recurrence time to $x_0$ or $G$. Third, despite its mathematical simplicity, the approach yields precise and explicit bounds on the corrections to exponentiality. (2) We compare and relate different metastability conditions proposed in the literature so to eliminate potential sources of confusion. This is specially relevant for evolutions of infinite-volume systems, whose treatment depends on whether and how relevant parameters (temperature, fields) are adjusted. (3) We introduce the notion of early asymptotic exponential behavior to control time scales asymptotically smaller than the mean-time scale. This control is particularly relevant for systems with unbounded state space where nucleations leading to exit from metastability can happen anywhere in the volume. We provide natural sufficient conditions on recurrence times for this early exponentiality to hold and show that it leads to estimations of probability density functions

Z', new fermions and flavor changing processes, constraints on E6_6 models from μ\mu --> eee

We study a new class of flavor changing interactions, which can arise in models based on extended gauge groups (rank $>$4) when new charged fermions are present together with a new neutral gauge boson. We discuss the cases in which the flavor changing couplings in the new neutral current coupled to the $Z^\prime$ are theoretically expected to be large, implying that the observed suppression of neutral flavor changing transitions must be provided by heavy $Z^\prime$ masses together with small $Z$-$Z^\prime$ mixing angles. Concentrating on E$_6$ models, we show how the tight experimental limit on $\mu \rightarrow eee$ implies serious constraints on the $Z^\prime$ mass and mixing angle. We conclude that if the value of the flavor changing parameters is assumed to lie in a theoretically natural range, in most cases the presence of a $Z^\prime$ much lighter than 1 TeV is unlikely.Comment: plain tex, 22 pages + 2 pages figures in PostScript (appended after `\bye'), UM-TH 92-1

Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary

In this paper we study the metastable behavior of the lattice gas in two and three dimensions subject to Kawasaki dynamics in the limit of low temperature and low density. We consider the local version of the model, where particles live on a finite box and are created, respectively, annihilated at the boundary of the box in a way that reflects an infinite gas reservoir. We are interested in how the system nucleates, i.e., how it reaches a full box when it starts from an empty box. Our approach combines geometric and potential theoretic arguments. In two dimensions, we identify the full geometry of the set of critical droplets for the nucleation, compute the average nucleation time up to a multiplicative factor that tends to one in the limit of low temperature and low density, express the proportionality constant in terms of certain capacities associated with simple random walk, and compute the asymptotic behavior of this constant as the system size tends to infinity. In three dimensions, we obtain similar results but with less control over the geometry and the constant. A special feature of Kawasaki dynamics is that in the metastable regime particles move along the border of a droplet more rapidly than they arrive from the boundary of the box. The geometry of the critical droplet and the sharp asymptotics for the average nucleation time are highly sensitive to this motion

Early mandibular canine-lateral incisor transposition: case report

Purpose. The main aim of the present study is to present a case of mandibular transposition between lateral incisor and canine in a paediatric patient. Materials and methods. A fixed multibracket orthodontic treatment was performed by means of a modified welded arch as to correct the transposition and obtaining a class I functional and symmetrical occlusion, also thanks to the early diagnosis of the eruption anomaly. Results. Our case report shows that a satisfactory treatment of mandibular transpositions is obtained when detected at an early stage of the tooth development. Conclusions. The main treatment options to be taken into consideration in case of a mandibular transposition are two: correcting the transposition or aligning it leaving the dental elements in their transposed order; in both cases, the followups show a stable condition, maintained without relapses. Several factors, such as age of the patient, occlusion, aesthetics, patient’s collaboration, periodontal support and duration of treatment have to be considered as to prevent potential damage to dental elements and support appliances. The choice between the two treatment approaches for mandibular lateral incisor/canine transpositions mainly depends on the time the anomaly is detected

Correlation between parodontal indexes and orthodontic retainers: prospective study in a group of 16 patients

Purpose. Fixed retainers are used to stabilize dental elements after orthodontic treatment. Being it a permanent treatment, it is necessary to instruct patients about a constant and continuous monitoring of their periodontal conditions and a correct oral hygiene. The aim of this study was to highlight the possible adverse effects of bonded retainers on parameters correlated to the health conditions of periodontal tissues. Materials and methods. We selected 16 patients, under treatment in the Orthodontics Department of University of Bari Dental School, who had undergone a lingual retainer insertion at the end of the orthodontic treatment. The patients were then divided into two groups (Control Group and Study Group) and monitored for 3 and 36 months, respectively. The following indexes were taken into consideration: gingival index (GI), plaque index (PI) and the presence of calculus (Calculus Index, CI), the probing depth and the presence of gingival recession on the six inferior frontal dental elements. Results. After the observation was carried out, any of the patients showed periodontal sockets and gingival recession. In the Study Group, only 1 patient had a PI score=3, the 7 left had scores between 0.66 and 2.83. In the Control Group, one patient had score=0, the other ones showed values between 0.5 and 1.66. The mean GI in the Study Group peaked at a score of 2.83, the minimum was 0.66; whereas in the Control Group the maximum value was 2 and the minimum 0.66. The CI in the Group Study was between 1 and 2. In the Control Group it was absent in only 1 patient, whereas in the remaining 7, it had a value between 0.3 and 1. The clinical data were studied by means of the Wilcoxon test. We found a statistically significant difference for what concerns the Plaque Indexes (PI) (P>0.05) and Calculus Indexes (CI) (P>0.1) in both groups, with higher scores in the Study Group, having retainers for 36 months. Any statistically significant difference was calculated for the GI. Conclusions. We can therefore conclude that patients with lingual retainers need periodontal hygiene and treatment as to prevent, in the course of time, periodontal damages non-detectable in short-term

Kawasaki dynamics with two types of particles: stable/metastable configurations and communication heights

This is the second in a series of three papers in which we study a two-dimensional lattice gas consisting of two types of particles subject to Kawasaki dynamics at low temperature in a large finite box with an open boundary. Each pair of particles occupying neighboring sites has a negative binding energy provided their types are different, while each particle has a positive activation energy that depends on its type. There is no binding energy between particles of the same type. At the boundary of the box particles are created and annihilated in a way that represents the presence of an infinite gas reservoir. We start the dynamics from the empty box and are interested in the transition time to the full box. This transition is triggered by a critical droplet appearing somewhere in the box. In the first paper we identified the parameter range for which the system is metastable, showed that the first entrance distribution on the set of critical droplets is uniform, computed the expected transition time up to and including a multiplicative factor of order one, and proved that the nucleation time divided by its expectation is exponentially distributed, all in the limit of low temperature. These results were proved under three hypotheses, and involve three model-dependent quantities: the energy, the shape and the number of critical droplets. In this second paper we prove the first and the second hypothesis and identify the energy of critical droplets. The paper deals with understanding the geometric properties of subcritical, critical and supercritical droplets, which are crucial in determining the metastable behavior of the system. The geometry turns out to be considerably more complex than for Kawasaki dynamics with one type of particle, for which an extensive literature exists. The main motivation behind our work is to understand metastability of multi- type particle systems.Comment: 31 pages, 22 figure

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

Long paths in first passage percolation on the complete graph I. Local PWIT dynamics

We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed edge weights. We find classes with different behaviour depending on a sequence of parameters (sn)n≥1 that quantifies the extreme-value behavior of small weights. We consider both n-independent as well as n-dependent edge weights and illustrate our results in many examples. In particular, we investigate the case where sn → ∞, and focus on the exploration process that grows the smallest-weight tree from a vertex. We establish that the smallest-weight tree process locally converges to the invasion percolation cluster on the Poisson-weighted infinite tree, and we identify the scaling limit of the weight of the smallest-weight path between two uniform vertices. In addition, we show that over a long time interval, the growth of the smallest-weight tree maintains the same volume-height scaling exponent – volume proportional to the square of the height – found in critical Galton–Watson branching trees and critical Erdős-Rényi random graphs