BBS invariant measures with independent soliton components

Ferrari, Pablo A.; Gabrielli, Davide

doi:10.48550/arxiv.1812.02437

Cited by 5 publications

(11 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recently there has been a renewed interest on BBS from the perspectives of statistical physics and probability theory in and out of equilibrium [6,7,8,18,19,29,30,25]. Our aim in this paper is to explore such features further in the light of generalized hydrodynamics (GHD).…”

Section: Arxiv:200401569v2 [Math-ph] 16 Apr 2020mentioning

confidence: 99%

Generalized hydrodynamics in box-ball system

Kuniba¹,

Misguich²,

Pasquier³

2020

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

Box-ball system (BBS) is a prominent example of integrable cellular automata in one dimension connected to quantum groups, Bethe ansatz, ultradiscretization, tropical geometry and so forth. In this paper we study the generalized Gibbs ensemble of BBS soliton gas by thermodynamic Bethe ansatz and generalized hydrodynamics. The results include the solution to the speed equation for solitons, an intriguing connection of the effective speed with the period matrix of the tropical Riemann theta function, an explicit description of the density plateaux that emerge from domain wall initial conditions including their diffusive corrections.

show abstract

Section: Arxiv:200401569v2 [Math-ph] 16 Apr 2020mentioning

confidence: 99%

Generalized hydrodynamics in box-ball system

Kuniba¹,

Misguich²,

Pasquier³

2020

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…We report here a family of distributions on the set of excursions proposed by the authors [6] based on the slot decomposition of the excursions. We include a theorem in the same paper which shows that the measure seen in the soliton components of the slot diagram are conditionally independent geometric random variables.…”

Section: Soliton Distributionmentioning

confidence: 99%

“…On the other hand, to complete the proof of (b) it suffices to show that Qα ∈ Q. The proof of this fact is more involved and can be found in [6].…”

Section: A Distribution On the Set Of Excursionsmentioning

confidence: 99%

“…A notable property proven by [7] is that the k-component of the configuration T η is a shift of the k-component of η, the amount shifted depending on the m-components for m > k. As a consequence, [7] prove the invariance with respect to the dynamics of some random configuration constructed by suitable shift-invariant probability measures on the distribution of the solitons on the slots. In [6] a special class of these measures is studied in detail. The papers [2,3] show a families of invariant measures for the BBS based on reversible Markov chains on {0, 1}.…”

Section: Introductionmentioning

confidence: 99%

“…In §5 we review some results of the authors [6] related with the distribution of the soliton decomposition of excursions. In particular, the soliton decomposition of a simple random walk consists on independent double-infinite vectors of iid geometric random variables.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Box-ball system: soliton and tree decomposition of excursions

Ferrari¹,

Gabrielli²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

We review combinatorial properties of solitons of the Box-Ball system introduced by Takahashi and Satsuma. 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 proposed a soliton decomposition of an excursion over the current minima of the walk representative of a ball configuration. Building on this approach, we propose a new soliton decomposition which is equivalent to the classical branch decomposition of the tree associated to the excursion. When the ball occupation numbers are independent Bernoulli variables of parameter λ < 1/2, the representative is a simple random walk with negative drift 2λ − 1 and infinitely many excursions. The soliton decomposition of that walk consists on independent double-infinite vectors of iid geometric random variables, property shared by the branch decomposition of the excursion trees of the random walk.

show abstract

Invariant measures for the box-ball system based on stationary Markov chains and periodic Gibbs measures

Croydon

Sasada

2019

Journal of Mathematical Physics

View full text Add to dashboard Cite

The box-ball system (BBS) is a simple model of soliton interaction introduced by Takahashi and Satsuma in the 1990s. Recent work of the authors, together with Tsuyoshi Kato and Satoshi Tsujimoto, derived various families of invariant measures for the BBS based on two-sided stationary Markov chains [4]. In this article, we survey the invariant measures that were presented in [4], and also introduce a family of new ones for periodic configurations that are expressed in terms of Gibbs measures. Moreover, we show that the former examples can be obtained as infinite volume limits of the latter. Another aspect of [4] was to describe scaling limits for the box-ball system; here, we review the results of [4], and also present scaling limits other than those that were covered there. One, the zigzag process has previously been observed in the context of queuing; another, a periodic version of the zigzag process, is apparently novel. Furthermore, we demonstrate that certain Palm measures associated with the stationary and periodic versions of the zigzag process yield natural invariant measures for the dynamics of corresponding versions of the ultra-discrete Toda lattice.

show abstract

BBS invariant measures with independent soliton components

Cited by 5 publications

References 7 publications

Generalized hydrodynamics in box-ball system

Generalized hydrodynamics in box-ball system

Box-ball system: soliton and tree decomposition of excursions

Invariant measures for the box-ball system based on stationary Markov chains and periodic Gibbs measures

Contact Info

Product

Resources

About