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Atsuo Kuniba

Publications and source records attributed to Atsuo Kuniba.

At least 19 recordsLinked to original sources
arXiv2026

Functional relations in renormalization group methods for a class of ordinary differential equations

We develop a renormalization group (RG)-based perturbation scheme for a class of ordinary differential equations, including first-order systems with semisimple or nilpotent linear parts, as well as scalar higher-order equations. The key observation is that the secular coefficients arising in naive perturbation theory satisfy an exact functional relation. This yields, in a unified manner, several fundamental features of the RG method: the renormalized amplitudes satisfy a closed functional relation with a group-like structure, the RG equation governing their slow dynamics is obtained directly, the absence of secular terms is ensured to all orders, and the relation between bare and renormalized amplitudes admits an explicit inversion. The results extend earlier ones for second-order scalar equations.

math-ph↗
arXiv2026

Multispecies inhomogeneous $t$-PushTASEP with general capacity

We study an $n$-species $t$-PushTASEP, an integrable long-range stochastic process, on a one-dimensional periodic lattice with inhomogeneities $x_1,\ldots,x_L$ and arbitrary capacity $l$ at each lattice site. The Markov matrix is identified with an alternating sum of commuting transfer matrices over all fundamental representations of $U_t(\widehat{sl}_{n+1})$. Stationary probabilities are expressed in a matrix product form involving a fusion of quantized corner transfer matrices for the strange five-vertex model introduced by Okado, Scrimshaw, and the second author. The resulting partition function, which serves as the normalization factor of the stationary probabilities, is obtained from the $l=1$ case by a finite plethystic substitution of length $l$.

math-ph↗
arXiv2025

Solutions of 3D Reflection Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver

We construct a new solution $(R,K)$ to the three-dimensional reflection equation, a boundary analogue of the tetrahedron equation. The $R$-operator is the one obtained by Sun, Terashima, Yagi, and the authors in 2024, involving four quantum dilogarithms with arguments in the $q$-Weyl algebra. The new $K$-operator similarly involves ten such quantum dilogarithms. Our approach is based on the quantum cluster algebra associated with the symmetric butterfly quiver on the wiring diagram of type C.

math.QA↗
arXiv2025

Multispecies inhomogeneous $t$-PushTASEP from antisymmetric fusion

We investigate the recently introduced inhomogeneous $n$-species $t$-PushTASEP, a long-range stochastic process on a periodic lattice. A Baxter-type formula is established, expressing the Markov matrix as an alternating sum of commuting transfer matrices over all the fundamental representations of $U_t(\widehat{sl}_{n+1})$. This superposition acts as an inclusion-exclusion principle, selectively extracting the sequential particle transitions characteristic of the PushTASEP, while canceling forbidden channels. The homogeneous specialization connects the PushTASEP to ASEP, showing that the two models share eigenstates and a common integrability structure.

math-ph↗
arXiv2025

Quantized six-vertex model on a torus

We study the integrability of the quantized six-vertex model with four parameters on a torus. It is a three-dimensional integrable lattice model in which a layer transfer matrix, depending on two spectral parameters associated with the homology cycles of the torus, can be defined not only on the square lattice but also on more general graphs. For a class of graphs that we call admissible, we establish the commutativity of the layer transfer matrices by introducing four types of tetrahedron equations and two types of inversion relations. Expanding in the spectral parameters yields a family of commuting quantum Hamiltonians. The quantized six-vertex model can also be reformulated in terms of (quantized) dimer models, and encompasses known integrable systems as special cases, including the free parafermion model and the relativistic Toda chain.

nlin.SI↗
arXiv2024

Solutions of Tetrahedron Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver

We construct a new solution to the tetrahedron equation by further pursuing the quantum cluster algebra approach in our previous works. The key ingredients include a symmetric butterfly quiver attached to the wiring diagrams for the longest element of type $A$ Weyl groups and the implementation of quantum $Y$-variables through the $q$-Weyl algebra. The solution consists of four products of quantum dilogarithms. By exploring both the coordinate and momentum representations, along with their modular double counterparts, our solution encompasses various known three-dimensional (3D) $R$-matrices. These include those obtained by Kapranov-Voevodsky (1994) utilizing the quantized coordinate ring, Bazhanov-Mangazeev-Sergeev (2010) from a quantum geometry perspective, Kuniba-Matsuike-Yoneyama (2023) linked with the quantized six-vertex model, and Inoue-Kuniba-Terashima (2023) associated with the Fock-Goncharov quiver. The 3D $R$-matrix presented in this paper offers a unified perspective on these existing solutions, coalescing them within the framework of quantum cluster algebra.

math.QA↗
arXiv2024

A strange five vertex model and multispecies ASEP on a ring

We revisit the problem of constructing the stationary states of the multispecies asymmetric simple exclusion process on a one-dimensional periodic lattice. Central to our approach is a quantum oscillator weighted five vertex model which features a strange weight conservation distinct from the conventional one. Our results clarify the interrelations among several known results and refine their derivations. For instance, the stationary probability derived from the multiline queue construction by Martin (2020) and Corteel--Mandelshtam--Williams (2022) is identified with the partition function of a three-dimensional system. The matrix product operators by Prolhac--Evans--Mallick (2009) acquire a natural diagrammatic interpretation as corner transfer matrices (CTM). The origin of their recursive tensor structure, as questioned by Aggarwal--Nicoletti--Petrov (2023), is revealed through the CTM diagrams. Finally, the derivation of the Zamolodchikov--Faddeev algebra by Cantini--de Gier--Wheeler (2015) is made intrinsic by elucidating its precise connection to a solution to the Yang--Baxter equation originating from quantum group representations.

math-ph↗
arXiv2024

Tetrahedron equation and quantum cluster algebras

We develop the quantum cluster algebra approach recently introduced by Sun and Yagi to investigate the tetrahedron equation, a three-dimensional generalization of the Yang-Baxter equation. In the case of square quiver, we devise a new realization of quantum Y-variables in terms $q$-Weyl algebras and obtain a solution that possesses three spectral parameters. It is expressed in various forms, comprising four products of quantum dilogarithms depending on the signs in decomposing the quantum mutations into the automorphism part and the monomial part. For a specific choice of them, our formula precisely reproduces Sergeev's $R$ matrix, which corresponds to a vertex formulation of the Zamolodchikov-Bazhanov-Baxter model when $q$ is specialized to a root of unity.

math.QA↗
arXiv2023

Quantum cluster algebras and 3D integrability: Tetrahedron and 3D reflection equations

We construct a new solution to the tetrahedron equation and the three-dimensional (3D) reflection equation by extending the quantum cluster algebra approach by Sun and Yagi concerning the former. We consider the Fock-Goncharov quivers associated with the longest elements of the Weyl groups of type $A$ and $C$, and investigate the cluster transformations corresponding to changing a reduced expression into a `most distant' one. By devising a new realization of the quantum $y$-variables in terms of $q$-Weyl algebra, the solutions are extracted as the operators whose adjoint actions yield the cluster transformations of the quantum $y$-variables. Explicit formulas of their matrix elements are also derived for some typical representations.

math.QA↗
arXiv2023

New solutions to the tetrahedron equation associated with quantized six-vertex models

We present a family of new solutions to the tetrahedron equation of the form $RLLL=LLLR$, where $L$ operator may be regarded as a quantized six-vertex model whose Boltzmann weights are specific representations of the $q$-oscillator or $q$-Weyl algebras. When the three $L$'s are associated with the $q$-oscillator algebra, $R$ coincides with the known intertwiner of the quantized coordinate ring $A_q(sl_3)$. On the other hand, $L$'s based on the $q$-Weyl algebra lead to new $R$'s whose elements are either factorized or expressed as a terminating $q$-hypergeometric type series.

math-ph↗
arXiv2022

Current correlations, Drude weights and large deviations in a box-ball system

We explore several aspects of the current fluctuations and correlations in the box-ball system (BBS), an integrable cellular automaton in one space dimension. The state we consider is an ensemble of microscopic configurations where the box occupancies are independent random variables (i.i.d. state), with a given mean ball density. We compute several quantities exactly in such homogeneous stationary state: the mean value and the variance of the number of balls $N_t$ crossing the origin during time $t$, and the scaled cumulants generating function associated to $N_t$. We also compute two spatially integrated current-current correlations. The first one, involving the long-time limit of the current-current correlations, is the so-called Drude weight and is obtained with thermodynamic Bethe Ansatz (TBA). The second one, involving equal time current-current correlations is calculated using a transfer matrix approach. A family of generalized currents, associated to the conserved charges and to the different time evolutions of the models are constructed. The long-time limits of their correlations generalize the Drude weight and the second cumulant of $N_t$ and are found to obey nontrivial symmetry relations. They are computed using TBA and the results are found to be in good agreement with microscopic simulations of the model. TBA is also used to compute explicitly the whole family of flux Jacobian matrices. Finally, some of these results are extended to a (non-i.i.d.) two-temperatures generalized Gibbs state (with one parameter coupled to the total number of balls, and another one coupled to the total number of solitons).

math-ph↗
arXiv2022

On tetrahedron type equations associated with $B_3,C_3, F_4$ and $H_3$

Tetrahedron equation is a three dimensional analogue of the Yang-Baxter equation. It allows a formulation in terms of the Coxeter group $A_3$. This short note includes miscellaneous remarks on the generalizations along $B_3, C_3, F_4$ and the non-crystallographic Coxeter group $H_3$. It is a supplement to the author's talk in the online workshop, Combinatorial Representation Theory and Connections with Related Fields, at RIMS, Kyoto University in November 2021.

math-ph↗
arXiv2021

A remark on renormalization group theoretical perturbation in a class of ordinary differential equations

We revisit the renormalization group (RG) theoretical perturbation theory on oscillator-type second-order ordinary differential equations. For a class of potentials, we show a simple functional relation among secular coefficients of the harmonics in the naive perturbation series. It leads to an inversion formula between bare and renormalized amplitudes and an elementary proof of the absence of secular terms in all orders of the RG series. The result covers nonautonomous as well as autonomous cases and refines earlier studies, including the classic examples of Van der Pol, Mathieu, Duffing, and Rayleigh equations.

math-ph↗
arXiv2021

Generalized hydrodynamics in complete box-ball system for $U_q(\widehat{sl}_n)$

We introduce the complete box-ball system (cBBS), which is an integrable cellular automaton on 1D lattice associated with the quantum group $U_q(\widehat{sl}_n)$. Compared with the conventional $(n-1)$-color BBS, it enjoys a remarkable simplification that scattering of solitons is totally diagonal. We also submit the cBBS to randomized initial conditions and study its non-equilibrium behavior by thermodynamic Bethe ansatz and generalized hydrodynamics. Excellent agreement is demonstrated between theoretical predictions and numerical simulation on the density plateaux generated from domain wall initial conditions including their diffusive broadening.

nlin.SI↗
arXiv2020

Generalized hydrodynamics in box-ball system

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.

math-ph↗
arXiv2019

Large deviations and one-sided scaling limit of randomized multicolor box-ball system

The basic $κ$-color box-ball (BBS) system is an integrable cellular automaton on one dimensional lattice whose local states take $\{0,1,\cdots,κ\}$ with $0$ regarded as an empty box. The time evolution is defined by a combinatorial rule of quantum group theoretical origin, and the complete set of conserved quantities is given by a $κ$-tuple of Young diagrams. In the randomized BBS, a probability distribution on $\{0,1,\cdots,κ\}$ to independently fill the consecutive $n$ sites in the initial state induces a highly nontrivial probability measure on the $κ$-tuple of those invariant Young diagrams. In a recent work \cite{kuniba2018randomized}, their large $n$ `equilibrium shape' has been determined in terms of Schur polynomials by a Markov chain method and also by a very different approach of Thermodynamic Bethe Ansatz (TBA). In this paper, we establish a large deviations principle for the row lengths of the invariant Young diagrams. As a corollary, they are shown to converge almost surely to the equilibrium shape at an exponential rate. We also refine the TBA analysis and obtain the exact scaling form of the vacancy, the row length and the column multiplicity, which exhibit nontrivial factorization in a one-parameter specialization.

math.PR↗
arXiv2019

Quantum spin chains from Onsager algebras and reflection $K$-matrices

We present a representation of the generalized $p$-Onsager algebras $O_p(A^{(1)}_{n-1})$, $O_p(D^{(2)}_{n+1})$, $O_p(B^{(1)}_n)$, $O_p(\tilde{B}^{(1)}_n)$ and $O_p(D^{(1)}_n)$ in which the generators are expressed as local Hamiltonians of XXZ type spin chains with various boundary terms reflecting the Dynkin diagrams. Their symmetry is described by the reflection $K$ matrices which are obtained recently by a $q$-boson matrix product construction related to the 3D integrability and characterized by Onsager coideals of quantum affine algebras. The spectral decomposition of the $K$ matrices with respect to the classical part of the Onsager algebra is described conjecturally. We also include a proof of a certain invariance property of boundary vectors in the $q$-boson Fock space playing a key role in the matrix product construction.

math-ph↗