Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1905.00186

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

David A. Croydon·Makiko Sasada
View original source ↗Read source document ↗Export metadata ↓

Abstract

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. In this article, we survey the invariant measures that were presented in the latter work, 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 the previous work was to describe scaling limits for the box-ball system; here, we review these results, 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.

Explore related subjects

math.PRmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

David A. Croydon, Makiko Sasada. 2019-05-01. Invariant measures for the box-ball system based on stationary Markov chains and periodic Gibbs measures. https://doi.org/10.1063/1.5095622

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Generalized Edgeworth expansions for integer-valued additive functionals of uniformly elliptic Markov chains

We obtain asymptotic expansions for probabilities $\bbP(S_N=k)$ of partial sums of uniformly bounded integer-valued functionals $\DS S_N=\sum_{n=1}^N f_n(X_n)$ of uniformly elliptic inhomogeneous Markov chains. The expansions involve products of polynomials and trigonometric polynomials, and they hold without additional assumptions. As an application of the explicit formulas of the trigonometric polynomials, we relate existence of the standard Edgeworth expansions of order $r$ to the rate of equidistributions of $S_N$ modulo $m$ for small positive integers $m.$

math.PR↗
arXiv2026

Permutations from Random Walk

Xavier and Yushi run a "random race" as follows. An atomless probability distribution $μ$ on the real line is chosen. The runners begin at zero. At time $i$ Xavier draws $\mathbf{X}_i$ from $μ$ and advances that distance, while Yushi advances by an independent drawing $\mathbf{Y}_i$. After $n$ such moves, what is the probability that Yushi led all the way? That the answer (namely, $4^{-n}\binom{2n}{n}$) is independent of $μ$ follows from a classical theorem of Darling, stating that for symmetric atomless increments, the distribution of each individual rank in the permutation obtained by ranking the partial sums is independent of the step law. We give a self-contained proof and extend the result to the permutations generated by partial sums of uniformly random signed permutations of any fixed, finite, generic set of reals. For atomless increments with mean zero and finite variance, without assuming symmetry, we show that random-walk permutations approach a random object that we call the "Wiener permuton," whose expected pattern densities equal the probabilities of the corresponding permutations generated by finite random walks with centered Laplace increments. Finally, we exhibit an infinite family of constructions whose limiting permutons interpolate between the Wiener permuton and the recursive separable permuton; each has the same intensity permuton, providing a single two-dimensional extension of the classical arcsine law for all of them.

math.PR↗
arXiv2026

On the uniqueness of quasi-stationary distributions for population models with spatial structure

Subcritical population processes are attracted to extinction and do not have non-trivial stationary distributions, which prompts the study of quasi-stationary distributions (QSDs) instead. In contrast to what generally happens for stationary distributions, QSDs may not be unique, even under irreducibility conditions. The general conditions for uniqueness of QSDs are not always easy to check. For the branching process, besides the quasi-limiting distribution there are many other QSDs. In this paper, we investigate whether adding little extra information to the continuous-time branching process is enough to obtain uniqueness. We consider the branching process with genealogy and branching random walks, and show that they have a unique QSD.

math.PR↗