Abstract
We study an infinite system of moving particles, where each particle is of type A or B. Particles perform independent random walks at rates \(D_A > 0\) and \(D_B \geqslant 0\), and the interaction is given by mutual annihilation \(A+B \rightarrow \emptyset \). The initial condition is i.i.d. with finite first moment. We show that this system is site-recurrent, that is, each site is visited infinitely many times. We also generalize a lower bound on the density decay of Bramson and Lebowitz by considering a construction that handles different jump rates.
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Models with mutual annihilation were originally introduced in chemical physics for the study of radiation-chemical processes in polymers. It was proposed in [19] that a radical may move along a polymer chain, and that the act of recombination takes place when two migrating radicals encounter one another. A diffusive mechanism for the motion of radicals was proposed in [22], and a kinetic equation describing concentration of free radicals as function of the time was derived. The same model served as a prototype of multi-type diffusion-limited chemical reactions with annihilation or inert compound outcome [10, 23], and as caricature modeling particle–antiparticle annihilation of superheavy magnetic monopoles in the very early universe [26].
Our motivation comes from the study of driven-dissipative lattice gases which undergo absorbing-state phase transitions. The authors arrived to the present model as a caricature of a system starting from an active state with critical density [12, 24]. The A-particles should correspond to regions that are slightly supercritical due to fluctuations, whereas B-particles represent slightly subcritical regions. Surprisingly enough, some of the techniques developed in this paper have been applied with success in the study of the original model [11].
In multi-type systems particles are of types \(A_1,A_2,\dots ,A_M\), and jump at rate 1 according to a generously transitive transition kernel \(p(\cdot ,\cdot )\). Interaction is given by \(A_i+A_j \rightarrow \emptyset \) for any \(i \ne j\). Each site initially contains one particle of type \(A_i\) with probability \(\frac{p}{M}\) and no particles with probability \(1-p\), independently of other sites. In the one-type system the interaction is given by \(A+A\rightarrow \emptyset \) and the initial condition is i.i.d. Bernoulli. The proofs given in the next sections for site recurrence work in these settings. For the one-type system, there is a simpler and more general proof in [4].
An event is “local” if its occurrence is determined by \(\big (\xi _t(x)\big )_{|x|\leqslant M,t\leqslant M}\) for some \(M<\infty \).
A graph G being generously transitive is stronger than being unimodular, and it is neither stronger nor weaker than being Cayley. An example of a graph that is generously transitive but not Cayley is the product \(P \times \mathbb {Z}\), where P is the Petersen graph. Cayley graphs of Abelian groups are generously transitive. An example of a graph that is not generously transitive but is Cayley is the free product \(\mathbb {Z}_2 * \mathbb {Z}_3\).
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Acknowledgements
We thank E. Andjel for fruitful discussions. M.C. and V.S thank MSRI for hospitality and support. This project was supported by grants Programa Iniciativa Científica Milenio grant number NC130062 through Nucleus Millennium Stochastic Models of Complex and Disordered Systems, PIP 11220130100521CO, PICT-2015-3154, PICT-2013-2137, PICT-2012-2744, Conicet-45955 and MinCyT-BR-13/14.
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Cabezas, M., Rolla, L.T. & Sidoravicius, V. Recurrence and density decay for diffusion-limited annihilating systems. Probab. Theory Relat. Fields 170, 587–615 (2018). https://doi.org/10.1007/s00440-017-0763-3
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DOI: https://doi.org/10.1007/s00440-017-0763-3