Box-ball system: soliton and tree decomposition of excursions

Ferrari, Pablo A.; Gabrielli, Davide

doi:10.48550/arxiv.1906.06405

Cited by 2 publications

(4 citation statements)

References 19 publications

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“…We have that E is in bijection with S so that a slot diagram completely codifies an excursion. We now construct the map ε → x[ε] and its inverse x → ε[x] (see [3] and [2] for more details).…”

Section: Excursions Solitons and Slot Diagramsmentioning

confidence: 99%

“…We have that x (i) i = −3, −1, 0, 1, 2 are the ones illustrated in Fig. 7 where x (−3) = x (2) = ∅. The integer labels below the picture represents the coordinates.…”

Section: Concatenation Of Slot Diagramsmentioning

confidence: 99%

“…The soliton decomposition of an infinite configuration of balls is obtained applying the TS algorithm independently to each single finite excursion. See also [2] for a different soliton decomposition related to the trees underlying excursions.…”

Section: Introductionmentioning

confidence: 99%

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BBS invariant measures with independent soliton components

Ferrari¹,

Gabrielli²

2018

Preprint

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The Box-Ball System (BBS) is a one-dimensional cellular automaton in {0, 1} Z introduced by Takahashi and Satsuma [7], who also identified conserved sequences called solitons. Integers are called boxes and a ball configuration indicates the boxes occupied by balls. For each integer k ≥ 1, a k-soliton consists of k boxes occupied by balls and k empty boxes (not necessarily consecutive). Ferrari, Nguyen, Rolla and Wang [3] define the k-slots of a configuration as the places where ksolitons can be inserted. Labeling the k-slots with integer numbers, they define the k-component of a configuration as the array {ζ k (j)} j∈Z of elements of Z ≥0 giving the number ζ k (j) of k-solitons appended to k-slot j ∈ Z. They also show that if the Palm transform of a translation invariant distribution µ has independent soliton components, then µ is invariant for the automaton. We show that for each λ ∈ [0, 1/2) the Palm transform of a product Bernoulli measure with parameter λ has independent soliton components and that its k-component is a product measure of geometric random variables with parameter 1 − q k (λ), an explicit function of λ. The construction is used to describe a large family of invariant measures with independent components under the Palm transformation, including Markov measures.

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Section: Excursions Solitons and Slot Diagramsmentioning

confidence: 99%

“…We have that x (i) i = −3, −1, 0, 1, 2 are the ones illustrated in Fig. 7 where x (−3) = x (2) = ∅. The integer labels below the picture represents the coordinates.…”

Section: Concatenation Of Slot Diagramsmentioning

confidence: 99%

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BBS invariant measures with independent soliton components

Ferrari¹,

Gabrielli²

2018

Preprint

Self Cite

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“…configurations, the authors' progress upon the latter topic is reported here. Whilst the study of classical integrable systems from a probabilistic perspective has hitherto been somewhat limited, we note that beyond the results discussed, there have been increasing efforts in this direction, see [11,12,13,20,21,22,23] for some of the interesting developments.…”

Section: Introductionmentioning

confidence: 99%

Discrete integrable systems and Pitman's transformation

Croydon¹,

Sasada²

2020

Preprint

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We survey recent work that relates Pitman's transformation to a variety of classical integrable systems, including the box-ball system, the ultra-discrete and discrete KdV equations, and the ultra-discrete and discrete Toda lattice equations. It is explained how this connection enables the dynamics of the integrable systems to be initiated from infinite configurations, which is important in the study of invariant measures. In the special case of spatially independent and identically distributed configurations, progress on the latter topic is also reported.

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Box-ball system: soliton and tree decomposition of excursions

Cited by 2 publications

References 19 publications

BBS invariant measures with independent soliton components

BBS invariant measures with independent soliton components

Discrete integrable systems and Pitman's transformation

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