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- W3100928556 abstract "We consider $(M,d)$ a connected and compact manifold and we denote by $mathcal{B}_i$ the Bernoulli space $M^{Z}$ of sequences represented by $$x=(... x_{-3},x_{-2},x_{-1},x_0,x_1,x_2,x_3,...),$$ where $x_i$ belongs to the space (alphabet) $M$. The case where $M=mathbb{S}^1$, the unit circle, is of particular interest here. The analogous problem in the one-dimensional lattice $mathbb{N}$ is also considered. %In this case we consider the potential $A: {cal B}=M^mathbb{N} to mathbb{R}.$ Let $A: mathcal{B}_i rar R$ be an {it observable} or {it potential} defined in the Bernoulli space $mathcal{B}_i$. The potential $A$ describes an interaction between sites in the one-dimensional lattice $M^mathbb{Z}$. Given a temperature $T$, we analyze the main properties of the Gibbs state $hat{mu}_{frac{1}{T} A}$ which is a certain probability measure over ${cal B}_i$. We denote this setting the general XY model. In order to do our analysis we consider the Ruelle operator associated to $frac{1}{T} A$, and, we get in this procedure the main eigenfunction $psi_{frac{1}{T} A}$. Later, we analyze selection problems when temperature goes to zero: a) existence, or not, of the limit (on the uniform convergence) $$V:=lim_{Tto 0} T, log(psi_{frac{1}{T} A}),,,,,text{a question about selection of subaction},$$ and, b) existence, or not, of the limit (on the weak$^*$ sense) $$tilde{mu}:=lim_{Tto 0} hat{mu}_{frac{1}{T}, A},,,,,text{a question about selection of measure}.$$ The existence of subactions and other properties of Ergodic Optimization are also considered." @default.
- W3100928556 created "2020-11-23" @default.
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- W3100928556 date "2011-11-01" @default.
- W3100928556 modified "2023-10-14" @default.
- W3100928556 title "ON THE GENERAL ONE-DIMENSIONAL XY MODEL: POSITIVE AND ZERO TEMPERATURE, SELECTION AND NON-SELECTION" @default.
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- W3100928556 doi "https://doi.org/10.1142/s0129055x11004527" @default.
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