Dissipative particle systems on expanders
arxiv(2024)
摘要
We consider a general framework for multi-type interacting particle systems
on graphs, where particles move one at a time by random walk steps, different
types may have different speeds, and may interact, possibly randomly, when they
meet. We study the equilibrium time of the process, by which we mean the number
of steps taken until no further interactions can occur. Under a rather general
framework, we obtain high probability upper and lower bounds on the equilibrium
time that match up to a constant factor and are of order nlog n if there are
order n vertices and particles. We also obtain similar results for the
balanced two-type annihilation model of chemical reactions; here, the balanced
case (equal density of types) does not fit into our general framework and makes
the analysis considerably more difficult. Our models do not admit any exact
solution as for integrable systems or the duality approach available for some
other particle systems, so we develop a variety of combinatorial tools for
comparing processes in the absence of monotonicity.
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