Explicit sufficient invariants for an interacting particle system

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<jats:p>We introduce a new class of interacting particle systems on a graph<jats:italic>G</jats:italic>. Suppose initially there are<jats:italic>N</jats:italic><jats:sub><jats:italic>i</jats:italic></jats:sub>(0) particles at each vertex<jats:italic>i</jats:italic>of<jats:italic>G</jats:italic>, and that the particles interact to form a Markov chain: at each instant two particles are chosen at random, and if these are at<jats:italic>adjacent</jats:italic>vertices of<jats:italic>G</jats:italic>, one particle jumps to the other particle's vertex, each with probability 1/2. The process<jats:bold><jats:italic>N</jats:italic></jats:bold>enters a death state after a finite time when all the particles are in some<jats:italic>independent</jats:italic>subset of the vertices of<jats:italic>G</jats:italic>, i.e. a set of vertices with no edges between any two of them. The problem is to find the distribution of the death state, η<jats:sub><jats:italic>i</jats:italic></jats:sub>=<jats:italic>N</jats:italic><jats:sub><jats:italic>i</jats:italic></jats:sub>(∞), as a function of<jats:italic>N</jats:italic><jats:sub><jats:italic>i</jats:italic></jats:sub>(0).</jats:p><jats:p>We are able to obtain, for some special graphs, the<jats:italic>limiting</jats:italic>distribution of<jats:italic>N</jats:italic><jats:sub><jats:italic>i</jats:italic></jats:sub>if the total number of particles<jats:italic>N</jats:italic>→ ∞ in such a way that the fraction,<jats:italic>N</jats:italic><jats:sub><jats:italic>i</jats:italic></jats:sub>(0)/<jats:italic>S</jats:italic>= ξ<jats:sub><jats:italic>i</jats:italic></jats:sub>, at each vertex is held fixed as<jats:italic>N</jats:italic>→ ∞. In particular we can obtain the limit law for the graph<jats:italic>S</jats:italic><jats:sub>2</jats:sub>, the two-leaf star which has three vertices and two edges.</jats:p>

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