Glauber Dynamics for the mean-field Potts Model

We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with $q\geq 3$ states and show that it undergoes a critical slowdown at an inverse-temperature $\beta_s(q)$ strictly lower than the critical $\beta_c(q)$ for uniqueness of the thermodynamic limit. The dynamical critical $\beta_s(q)$ is the spinodal point marking the onset of metastability. We prove that when $\beta<\beta_s(q)$ the mixing time is asymptotically $C(\beta, q) n \log n$ and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order $n$. At $\beta=\beta_s(q)$ the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order $n^{4/3}$. For $\beta>\beta_s(q)$ the mixing time is exponentially large in $n$. Furthermore, as $\beta \uparrow \beta_s$ with $n$, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of $O(n^{-2/3})$ around $\beta_s$. These results form the first complete analysis of mixing around the critical dynamical temperature --- including the critical power law --- for a model with a first order phase transition.Comment: 45 pages, 5 figure

Estrutura populacional de Petrolisthes armatus (Gibbes) (Decapoda, Anomura, Porcellanidae) da Ilha do Farol, Matinhos, Paraná, Brasil Population structure of Petrolisthes armatus (Gibbes) (Decapoda, Anomura, Porcellanidae) from Farol Isle, Matinhos, Paraná, Brazil

<abstract language="eng">Population structure of the porcelain crab Petrolisthes armatus (Gibbes, 1850) from the rocky shore of the Farol isle (25º51'S, 48º32'W) was carried out in the following aspects: annual fluctuation of the density, distribution of the developmental stages and sexes over year round, size composition and recruiting period of juveniles into the population. Monthly collections were made in the three sites of the midlittoral, from November/89 to April/91, with the aid of a trap. The air temperature ranged from 13ºC to 28ºC, that of the surface water from 16ºC lo 29ºC and the salinity of the surface water from 06.7‰ to 31.5‰. A total of 5.585 individuals of Petrolisthes armatus was collected, among which, 116 were in the megalopa stage, 1,579 juveniles, 830 immature males, 729 immature females, 1,161 mature males, 744 mature females and 426 ovigerous females. The population density ranged from 15.3 to 305.0 ind.m-2, with a peak in January. This density is the highest among the porcelain crab populations known from the literature. The species has continuous reproduction, but, there were two main cohorts in the year: summer one that was longer and more abundant and the winter one that was shorter and the less numerous