Abstract
We introduce the “mass migration process” (MMP), a conservative particle system on ${\mathbb N}^{{\mathbb Z}^d}$. It consists in jumps of $k$ particles ($k\geq 1$) between sites, with a jump rate depending only on the state of the system at the departure and arrival sites of the jump. It generalizes misanthropes processes, hence zero range and target processes. After the construction of MMP, our main focus is on its invariant measures. We derive necessary and sufficient conditions for the existence of translation invariant and invariant product probability measures. In the particular cases of asymmetric mass migration zero range and mass migration target dynamics, these conditions yield explicit solutions. If these processes are moreover attractive, we obtain a full characterization of all translation invariant, invariant probability measures. We also consider attractiveness properties (through couplings), condensation phenomena, and their links for MMP. We illustrate our results on many examples; we prove the coexistence of condensation and attractiveness in one of them.
Highlights
In the study of an interacting particle system, an essential tool is the explicit knowledge of an invariant measure
Under conditions on the rates, this is the case for misanthropes processes, which include the more recently studied target processes [8, 15, 21]. All these dynamics consist in individual jumps of particles between sites, with rates which are the product of two terms: a transition probability giving the direction of the jump, and a function depending on the occupation numbers at the departure and/or arrival sites of the jump
In this paper we address this question for a class of models generalizing them, that we call mass migration processes
Summary
In the study of an interacting particle system, an essential tool is the explicit knowledge of an invariant measure. The MMP is a particular case of the multiple particle jump model studied in [14] in the context of attractiveness properties; under some conditions on the rates, the latter possesses a one parameter family of translation invariant and invariant measures, that are not always explicit. In the context of exactly solvable models and duality, the q-Hahn asymmetric zero range processes are other dynamics with zero range interaction and multiple jumps (see [3] and references therein) with explicit product invariant measures; this knowledge is crucial for exact solvability Another possible extension of these models are processes in which multiple births and deaths are superimposed to multiple jumps; their attractiveness properties have been established in [5], and they have been applied to the study of survival and extinction of species in [6].
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