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Register a new userSharp thresholds for Gibbs-non-Gibbs transition in the fuzzy Potts model with a Kac-type interaction
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We investigate the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac interaction. We use a variational approach for profiles inspired by that of Fernandez, den Hollander and Mart{i}nez for their study of the Gibbs-non-Gibbs transitions of a dynamical Kac-Ising model on the torus. As our main result, we show that the mean-field thresholds dividing Gibbsian from non-Gibbsian behavior are sharp in the fuzzy Kac-Potts model with class size unequal two. On the way to this result we prove a large deviation principle for color profiles with diluted total mass densities and use monotocity arguments.
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We continue our study of the full set of translation-invariant splitting Gibbs measures (TISGMs, translation-invariant tree-indexed Markov chains) for the $q$-state Potts model on a Cayley tree. In our previous work cite{KRK} we gave a full description of the TISGMs, and showed in particular that at sufficiently low temperatures their number is $2^{q}-1$. In this paper we find some regions for the temperature parameter ensuring that a given TISGM is (non-)extreme in the set of all Gibbs measures. In particular we show the existence of a temperature interval for which there are at least $2^{q-1} + q$ extremal TISGMs. For the Cayley tree of order two we give explicit formulae and some numerical values.
Gibbs-type random probability measures and the exchangeable random partitions they induce represent an important framework both from a theoretical and applied point of view. In the present paper, motivated by species sampling problems, we investigate some properties concerning the conditional distribution of the number of blocks with a certain frequency generated by Gibbs-type random partitions. The general results are then specialized to three noteworthy examples yielding completely explicit expressions of their distributions, moments and asymptotic behaviors. Such expressions can be interpreted as Bayesian nonparametric estimators of the rare species variety and their performance is tested on some real genomic data.
We consider an SOS (solid-on-solid) model, with spin values from the set of all integers, on a Cayley tree of order k and are interested in translation-invariant gradient Gibbs measures (GGMs) of the model. Such a measure corresponds to a boundary law (a function defined on vertices of the Cayley tree) satisfying a functional equation. In the ferromagnetic SOS case on the binary tree we find up to five solutions to a class of 4-periodic boundary law equations (in particular, some 2-periodic ones). We show that these boundary laws define up to four distinct GGMs. Moreover, we construct some 3-periodic boundary laws on the Cayley tree of arbitrary order k, which define GGMs different from the 4-periodic ones.
We show that nontrivial bi-infinite polymer Gibbs measures do not exist in typical environments in the inverse-gamma (or log-gamma) directed polymer model on the planar square lattice. The precise technical result is that, except for measures supported on straight-line paths, such Gibbs measures do not exist in almost every environment when the weights are independent and identically distributed inverse-gamma random variables. The proof proceeds by showing that when two endpoints of a point-to-point polymer distribution are taken to infinity in opposite directions but not parallel to lattice directions, the midpoint of the polymer path escapes. The proof is based on couplings, planar comparison arguments, and a recently discovered joint distribution of Busemann functions.
We study epidemic spreading according to a emph{Susceptible-Infectious-Recovered} (for short, emph{SIR}) network model known as the {em Reed-Frost} model, and we establish sharp thresholds for two generative models of {em one-dimensional small-world graphs}, in which graphs are obtained by adding random edges to a cycle. In $3$-regular graphs obtained as the union of a cycle and a random perfect matching, we show that there is a sharp threshold at $.5$ for the contagion probability along edges. In graphs obtained as the union of a cycle and of a $mathcal{G}_{n,c/n}$ ErdH{o}s-Renyi random graph with edge probability $c/n$, we show that there is a sharp threshold $p_c$ for the contagion probability: the value of $p_c$ turns out to be $sqrt 2 -1approx .41$ for the sparse case $c=1$ yielding an expected node degree similar to the random $3$-regular graphs above. In both models, below the threshold we prove that the infection only affects $mathcal{O}(log n)$ nodes, and that above the threshold it affects $Omega(n)$ nodes. These are the first fully rigorous results establishing a phase transition for SIR models (and equivalent percolation problems) in small-world graphs. Although one-dimensional small-world graphs are an idealized and unrealistic network model, a number of realistic qualitative phenomena emerge from our analysis, including the spread of the disease through a sequence of local outbreaks, the danger posed by random connections, and the effect of super-spreader events.
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