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Exact connections between current fluctuations and the second class particle in a class of deposition models
Comments: Second version, results a bit more clear; 23 pagesJournal: Journal of Stat. Phys., Volume 127, Number 2 / April, 2007Categories: math.PRKeywords: second class particle, exact connections, deposition models, attractive stochastic interacting systems, neighbor attractive stochastic interactingTags: journal articleWe consider a large class of nearest neighbor attractive stochastic interacting systems that includes the asymmetric simple exclusion, zero range, bricklayers' and the symmetric K-exclusion processes. We provide exact formulas that connect particle flux (or surface growth) fluctuations to the two-point function of the process and to the motion of the second class particle. Such connections have only been available for simple exclusion where they were of great use in particle current fluctuation investigations.
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Between equilibrium fluctuations and Eulerian scaling: Perturbation of equilibrium for a class of deposition models
Comments: 30 pages version 2: some typos corrected, some remarks addedWe investigate propagation of perturbations of equilibrium states for a wide class of 1D interacting particle systems. The class of systems considered incorporates zero range, $K$-exclusion, mysanthropic, `bricklayers' models, and much more. We do not assume attractivity of the interactions. We apply Yau's relative entropy method rather than coupling arguments. The result is \emph{partial extension} of T. Sepp\"al\"ainen's recent paper. For $0<\beta<1/5$ fixed, we prove that, rescaling microscopic space and time by $N$, respectively $N^{1+\beta}$, the macroscopic evolution of perturbations of microscopic order $N^{-\beta}$ of the equilibrium states is governed by Burgers' equation. The same statement should hold for $0<\beta<1/2$ as in Sepp\"al\"ainen's cited paper, but our method does not seem to work for $\beta\ge1/5$.
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Growth fluctuations in a class of deposition models
Comments: A minor mistake in lemma 5.1 is now correctedJournal: ALHP PR 39, 4 (2003) pp639-685Categories: math.PRKeywords: deposition models, growth fluctuations, second class particle, zero range processes, nearest-neighbor particle jump processesTags: journal articleWe compute the growth fluctuations in equilibrium of a wide class of deposition models. These models also serve as general frame to several nearest-neighbor particle jump processes, e.g. the simple exclusion or the zero range process, where our result turns to current fluctuations of the particles. We use martingale technique and coupling methods to show that, rescaled by time, the variance of the growth as seen by a deterministic moving observer has the form |V-C|*D, where V and C is the speed of the observer and the second class particle, respectively, and D is a constant connected to the equilibrium distribution of the model. Our main result is a generalization of Ferrari and Fontes' result for simple exclusion process. Law of large numbers and central limit theorem are also proven. We need some properties of the motion of the second class particle, which are known for simple exclusion and are partly known for zero range processes, and which are proven here for a type of deposition models and also for a type of zero range processes.