Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Box-ball system: soliton and tree decomposition of excursions   We review combinatorial properties of solitons of the Box-Ball system
introduced by Takahashi and Satsuma in 1990. Starting with several definitions
of the system, we describe ways to identify solitons and review a proof of the
conservation of the solitons under the dynamics. Ferrari, Nguyen, Rolla and
Wang 2018 proposed a soliton decomposition of a configuration into a family of
vectors, one for each soliton size. Based on this decompositions, the authors
have proposed a family of measures on the set of excursions which induces
invariant distributions for the Box-Ball System. In this paper, we propose a
new soliton decomposition which is equivalent to a branch decomposition of the
tree associated to the excursion, see Le Gall 2005. A ball configuration
distributed as independent Bernoulli variables of parameter $\lambda<1/2$ is in
correspondence with a simple random walk with negative drift $2\lambda-1$ and
infinitely many excursions over the local minima. In this case the authors have
proven that the soliton decomposition of the walk consists on independent
double-infinite vectors of iid geometric random variables. We show that this
property is shared by the branch decomposition of the excursion trees of the
random walk and discuss a corresponding construction of a Geometric branching
process with independent but not identically distributed Geometric random
variables.

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The categories of this paper are: plasm-ph
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The categories of this paper are: solv-int
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The categories of this paper are: nlin
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The categories of this paper are: bayes-an
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The categories of this paper are: adap-org
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The categories of this paper are: quant-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: physics
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The categories of this paper are: funct-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-alg
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The categories of this paper are: econ
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The categories of this paper are: math-ph