Periodic Box-ball System Research Articles

Geometric lifting of the integrable cellular automata with periodic boundary conditions

Inspired by G Frieden’s recent work on the geometric R-matrix for affine type A crystal associated with rectangular shaped Young tableaux, we propose a method to construct a novel family of discrete integrable systems which can be regarded as a geometric lifting of the generalized periodic box–ball systems. By converting the conventional usage of the matrices for defining the Lax representation of the discrete periodic Toda chain, together with a clever use of the Perron–Frobenious theorem, we give a definition of our systems. It is carried out on the space of real positive dependent variables, without regarding them to be written by subtraction-free rational functions of independent variables but nevertheless with the conserved quantities which can be tropicalized. We prove that, in this setup an equation of an analogue of the ‘carrier’ of the box–ball system for assuring its periodic boundary condition always has a unique solution. As a result, any states in our systems admit a commuting family of time evolutions associated with any rectangular shaped tableaux, in contrast to the case of corresponding generalized periodic box–ball systems where some states did not admit some of such time evolutions.

A geometric realization of the periodic discrete Toda lattice and its tropicalization

An explicit formula concerning curve intersections equivalent to the time evolution of the periodic discrete Toda lattice (pdTL) is presented. First, the time evolution is realized as a point addition on a hyperelliptic curve, which is the spectral curve of the pdTL, then the point addition is translated into curve intersections. Next, it is shown that the curves which appear in the curve intersections are explicitly given by using the conserved quantities of the pdTL. Finally, the formulation is lifted to the framework of tropical geometry and a tropical geometric realization of the periodic box–ball system is constructed via tropical curve intersections.

The two-dimensional periodic box–ball system and its fundamental cycle

We study a two-dimensional box–ball system which is an ultradiscrete analogue of the discrete KP equation. We construct an algorithm to calculate the fundamental cycle, which is an important conserved quantity of the two-dimensional box–ball system with periodic boundary condition, by using the tropical curve theory.

Two Point Correlation Functions for a Periodic Box-Ball System

We investigate correlation functions in a periodic box-ball system. For the second and the third nearest neighbor correlation functions, we give explicit formulae obtained by combinatorial methods. A recursion formula for a specific $N$-point functions is also presented.

Fermionic partition functions for a periodic soliton cellular automaton

Fermionic formulas in a combinatorial Bethe ansatz consist of sums of products of q-binomial coefficients. There exist refinements without a sum that are known to yield partition functions of box–ball systems with a prescribed soliton content. In this paper, such a refined fermionic formula is extended to the periodic box–ball system and a q-analog of the Bethe root counting formula for the XXZ chain at Δ = ∞.

Correlation functions for a periodic box–ball system

We investigate correlation functions in a periodic box–ball system. For the two-point functions of short distance, we give explicit formulae obtained by combinatorial methods. We give expressions for general N-point functions in terms of ultradiscrete theta functions.

Bethe Ansatz, Inverse Scattering Transform and Tropical Riemann Theta Function in a Periodic Soliton Cellular Automaton for An(1)

We study an integrable vertex model with a periodic boundary condition associated with U_q(A_n^{(1)}) at the crystallizing point q=0. It is an (n+1)-state cellular automaton describing the factorized scattering of solitons. The dynamics originates in the commuting family of fusion transfer matrices and generalizes the ultradiscrete Toda/KP flow corresponding to the periodic box-ball system. Combining Bethe ansatz and crystal theory in quantum group, we develop an inverse scattering/spectral formalism and solve the initial value problem based on several conjectures. The action-angle variables are constructed representing the amplitudes and phases of solitons. By the direct and inverse scattering maps, separation of variables into solitons is achieved and nonlinear dynamics is transformed into a straight motion on a tropical analogue of the Jacobi variety. We decompose the level set into connected components under the commuting family of time evolutions and identify each of them with the set of integer points on a torus. The weight multiplicity formula derived from the q=0 Bethe equation acquires an elegant interpretation as the volume of the phase space expressed by the size and multiplicity of these tori. The dynamical period is determined as an explicit arithmetical function of the n-tuple of Young diagrams specifying the level set. The inverse map, i.e., tropical Jacobi inversion is expressed in terms of a tropical Riemann theta function associated with the Bethe ansatz data. As an application, time average of some local variable is calculated.

Relationships Between Two Approaches: Rigged Configurations and 10-Eliminations

There are two distinct approaches to the study of initial value problem of the periodic box-ball systems. One way is the rigged configuration approach due to Kuniba--Takagi--Takenouchi and another way is the 10-elimination approach due to Mada--Idzumi--Tokihiro. In this paper, we describe precisely interrelations between these two approaches.

A crystal theoretic method for finding rigged configurations from paths

The Kerov–Kirillov–Reshetikhin (KKR) bijection gives one-to-one correspondences between the set of highest paths and the set of rigged configurations. In this paper, we give a crystal theoretic reformulation of the KKR map from the paths to rigged configurations, using the combinatorial R and energy functions. This formalism provides a tool for analysis of the periodic box-ball systems.

The box–ball system and the N-soliton solution of the ultradiscrete KdV equation

Any state of the box–ball system (BBS) together with its time evolution is described by the N-soliton solution (with appropriate choice of N) of the ultradiscrete KdV equation. It is shown that simultaneous elimination of all ‘10’-walls in a state of the BBS corresponds exactly to reducing the parameters that determine ‘the size of a soliton’ by one. This observation leads to an expression for the solution to the initial-value problem (IVP) for the BBS. Expressions for the solution to the IVP for the ultradiscrete Toda molecule equation and the periodic BBS are also presented.

Ultradiscretization of the theta function solution of pd Toda

A periodic box-ball system (pBBS) is obtained by ultradiscretizing the periodic discrete Toda equation (pd Toda equation). We show the relation between a Young diagram of the pBBS and a spectral curve of the pd Toda equation. The formula for the fundamental cycle of the pBBS is obtained as a corollary.

On the initial value problem of a periodic box-ball system

We show that the initial value problem of a periodic box-ball system can be solved in an elementary way using simple combinatorial methods.

The Bethe ansatz in a periodic box–ball system and the ultradiscrete Riemann thetafunction

Vertex models with quantum group symmetry give rise to integrable cellular automata atq = 0. We study a prototype example known as the periodic box–ball system. Theinitial value problem is solved in terms of an ultradiscrete analogue of theRiemann theta function whose period matrix originates in the Bethe ansatz atq = 0.

Bethe ansatz and inverse scattering transform in a periodic box–ball system

We formulate the inverse scattering method for a periodic box–ball system and solve the initial value problem. It is done by a synthesis of the combinatorial Bethe ansatzes at q = 1 and q = 0 , which provides the ultradiscrete analogue of quasi-periodic solutions in soliton equations, e.g., action–angle variables, Jacobi varieties, period matrices and so forth. As an application we establish explicit formulas counting the states characterized by conserved quantities and the generic and fundamental period under the commuting family of time evolutions.

The exact correspondence between conserved quantities of a periodic box-ball system and string solutions of the Bethe ansatz equations

We investigate the link between a periodic box-ball system (PBBS) and a solvable lattice model. Introducing a PBBS with an integer parameter corresponding to the dimensionality of the auxiliary space for the lattice model, we prove an important relationship between the conserved quantities of states of the PBBS and eigenvectors constructed through the string hypothesis.

Fundamental cycle of a periodic box–ball system and solvable lattice models

We investigate the fundamental cycle of a periodic box–ball system (PBBS) from a relation between the PBBS and a solvable lattice model. We show that the fundamental cycle of the PBBS is obtained from eigenvalues of the transfer matrix of the solvable lattice model.

FUNDAMENTAL CYCLE OF A PERIODIC BOX-BALL SYSTEM: A NUMBER THEORETICAL ASPECT

A number theoretical aspect of the fundamental cycle of a periodic box-ball system is investigated. Using the formulae for the fundamental cycle of a class of initial states, we point out that the asymptotic behaviour of the fundamental cycle is closely related to the celebrated Riemann hypothesis.

Path description of conserved quantities of generalized periodic box-ball systems

We investigate conserved quantities of periodic box-ball systems (PBBS) with arbitrary kinds of balls and box capacity greater than or equal to 1. We introduce the notion of nonintersecting paths on the two dimensional array of boxes, and give a combinatorial formula for the conserved quantities of the generalized PBBS using these paths.

Conserved quantities of generalized periodic box–ball systems constructed from the ndKP equation

We investigate periodic box–ball systems (PBBSs) with several kinds of balls and box capacity greater than or equal to one. Conserved quantities of the PBBSs are constructed from those of the nonautonomous discrete KP (ndKP) equation using the Lax representation of the ndKP equation.

Asymptotic behaviour of fundamental cycle of periodic box–ball systems

We investigate asymptotic behaviour of fundamental cycle of periodic box–ball systems (PBBSs) when the system size N goes to infinity. According to integrable nature of the PBBS, the trajectory is confined to qualitatively smaller number of states than that of the total states. We prove that, although the maximum fundamental cycle is of order of exp[√N], almost all fundamental cycle is less than exp[(logN)2].