Randomized box–ball systems, limit shape of rigged configurations and thermodynamic Bethe ansatz

Kuniba, Atsuo; Lyu, Hanbaek; Okado, Masato

doi:10.1016/j.nuclphysb.2018.10.008

Cited by 11 publications

(23 citation statements)

References 38 publications

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“…We remark that Z (1) 1 = p 0 + p 1 + · · · + p κ = 1 and π (1) c = p. Our proof of the irreducibility depends on an algorithmic characterization of the combinatorial R acting on B (a) c × B (1) 1 . On the other hand, the stationarity of π (a) c is shown by a similar argument for a more general result [14,Prop. 3.2] With Theorem 1 at hand, we can apply standard limit theorems for additive functionals of Markov chains for the row transfer matrix energy E (a) c (X n,p ).…”

Section: Statement Of Resultssupporting

confidence: 64%

“…In the last section, Section 6, we prove Theorem 7 using the Fermionic formula, and obtain a simple product formula for the limiting shape of the invariant Young diagrams (Theorem 8) by the method of Thermodynamic Bethe Ansatz. A more general result in this direction has been obtained recently in [14]. The case treated in this paper corresponds to its one parameter specialization, which deserves an independent report due to the nontrivial factorization of the final result in (31).…”

Section: Introductionsupporting

confidence: 52%

“…Now we are ready to prove Theorem 1. We note that the stationarity of π (a) c has been established in the more general BBS in the recent work [14,Prop. 3.2].…”

Section: 1mentioning

confidence: 56%

“…In order to make it a time evolution that preserves number of balls, we need to introduce 'barriers' at the right tail of the state space. See [14,Subsection 2.2] .…”

Section: The Carrier Processes and The Energy Matrixmentioning

confidence: 99%

“…A more general result has been given in Example 3.3 and equation (22) in [14], where ε (a) c is denoted by h (a) c .…”

Section: Theoremmentioning

confidence: 99%

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Large Deviations and One-Sided Scaling Limit of Randomized Multicolor Box-Ball System

Kuniba

Lyu

2019

J Stat Phys

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A random box-ball system starts with occupying each of the first n boxes independently with a ball of random color from {0, 1, · · · , κ}, where balls of color 0 are considered as empty boxes. The time evolution is defined by a successive application of the combinatorial R, and possesses a κ-tuple of Young diagrams as the complete set of conserved quantities. Using a Markov chain method, we show that if we scale the rows of each of the invariant Young diagrams by 1/n, it converges to some limiting shape as n → ∞ at an exponential rate. Furthermore, we determine the limiting shape by ratios of Schur polynomials with initial ball densities as parameters. We also derive similar results through an alternative method using the Fermionic form and Thermodynamic Bethe Ansatz, which apply once we condition the initial measure on the set of highest states. By a large deviations principle, we identify the limiting shapes of invariant Young diagrams corresponding to the unconditioned and conditioned initial measures.2010 Mathematics Subject Classification. 37K40, 60J10, 60F10.

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Section: Statement Of Resultssupporting

confidence: 64%

Section: Introductionsupporting

confidence: 52%

“…Now we are ready to prove Theorem 1. We note that the stationarity of π (a) c has been established in the more general BBS in the recent work [14,Prop. 3.2].…”

Section: 1mentioning

confidence: 56%

“…In order to make it a time evolution that preserves number of balls, we need to introduce 'barriers' at the right tail of the state space. See [14,Subsection 2.2] .…”

Section: The Carrier Processes and The Energy Matrixmentioning

confidence: 99%

“…A more general result has been given in Example 3.3 and equation (22) in [14], where ε (a) c is denoted by h (a) c .…”

Section: Theoremmentioning

confidence: 99%

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