Session S04 - Random Walks and Related Topics

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The Flip-Murder process exhibits a kind of first order phase transition

Alex Ramos

Federal University of Pernambuco, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

On the theory of interacting particle systems, the supposition that the set of particles does not change in the process of interaction is lore crystallized. This supposition, which we call constant-length, is not the only possible one. A. Toom and V. Malyshev[1,2] have considered another approximation, which we call variable-length. Here, we shall examine a random process motivated by this new paradigm. The process we have studied here and called Flip-Murder, it has discrete-time and its states are bi-infinite sequences, whose particles take only two values, denoted here as minus and plus. Our operator is a composition of the following two operators. The first operator, called flip, turns every minus into plus with probability $\beta$ independently from what happens at other places. The second operator, called murder, acts in the following way: whenever a plus is a left neighbor of a minus, this plus disappears with probability $\alpha$ independently from what happens at other places. We prove that our process exhibits regions of ergodicity and non-ergodicity on the parameter space; a kind of first order phase transition and an invariant measure whose density of plus is less than one. Moreover, we have performed some numerical investigations.

[1] MALYSHEV, V. A. Quantum grammars. Journal of Mathematical Physics, v.41, n.7, p. 4508-4520, 2000.

[2] TOOM, A. Non-ergodicity in a 1-D particle process with variable length. Journal of Statistical Physics, v. 115, n. 3-4, p. 895-924, 2004.

Joint work with L. T. Costa(Rural Federal University of Pernambuco).

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