Abstract
This study in centered on models accounting for stochastic deformations of sample paths of random walks,embedded either in Z 2 or in Z 3 . These models are immersed in multi-type particle systems with exclusion. Starting from examples, we give necessary and sufficient conditions for the underlying Markov processes to be reversible, in which case their invariant measure has a Gibbs form. Letting the size of the sample path increase, we find the convenient scalings bringing to light phase transition phenomena. Stable and metastable configurations are bound to time-periods of limiting deterministic trajectories which are solution of nonlinear differential systems: in the example of the ABC model, a system of Lotka-Volterra class is obtained, and the periods involve elliptic, hyper-elliptic or more general functions. Lastly, we discuss briefly the contour of a general approach allowing to tackle the transient regime via differential equations of Burgers’ type.
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Fayolle, G., Furtlehner, C. (2004). Stochastic Deformations of Sample Paths of Random Walks and Exclusion Models. In: Drmota, M., Flajolet, P., Gardy, D., Gittenberger, B. (eds) Mathematics and Computer Science III. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7915-6_41
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DOI: https://doi.org/10.1007/978-3-0348-7915-6_41
Publisher Name: Birkhäuser, Basel
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