The Functional Method for the Domain-Wall Partition Function
We review the (algebraic-)functional method devised by Galleas and further developed by Galleas and the author. We first explain the method using the simplest example: the computation of the partition function for the six-vertex model with domain-wall boundary conditions. At the heart of the method...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2018 |
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| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2018
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/209786 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | The Functional Method for the Domain-Wall Partition Function / J. Lamers // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 45 назв. — англ. |
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Lamers, J. 2025-11-26T12:23:34Z 2018 The Functional Method for the Domain-Wall Partition Function / J. Lamers // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 45 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 82B23; 30D05 arXiv: 1801.09635 https://nasplib.isofts.kiev.ua/handle/123456789/209786 https://doi.org/10.3842/SIGMA.2018.064 We review the (algebraic-)functional method devised by Galleas and further developed by Galleas and the author. We first explain the method using the simplest example: the computation of the partition function for the six-vertex model with domain-wall boundary conditions. At the heart of the method lies a linear functional equation for the partition function. After deriving this equation, we outline its analysis. The result is a closed expression in the form of a symmetrized sum - or, equivalently, multiple-integral formula - that can be rewritten to recover Izergin's determinant. Special attention is paid to the relation with other approaches. In particular, we show that the Korepin-Izergin approach can be recovered within the functional method. We comment on the functional method's range of applicability, and review how it is adapted to the technically more involved example of the elliptic solid-on-solid model with domain walls and a reflecting end. We present a new formula for the partition function of the latter, which was expressed as a determinant by Tsuchiya-Filali-Kitanine. Our result takes the form of a "crossing-symmetrized" sum with 2^L terms featuring the elliptic domain-wall partition function, which appears to be new also in the limiting case of the six-vertex model. Further taking the rational limit, we recover the expression obtained by Frassek using the boundary perimeter Bethe ansatz. This text has grown out of a presentation delivered at the Les Houches Summer School on Integrability in June 2016 and a poster presentation at the ESI Workshop Elliptic Hypergeometric Functions in Combinatorics, Integrable Systems and Physics in March 2017. It is a pleasure to thank the organizers of these excellent meetings. I am grateful to W. Galleas for collaboration on [20] and for suggesting the problem addressed in [31]. I thank G. Arutyunov and A. Henriques for discussions while working on [31], H. Rosengren for detailed feedback on [32] and the present text, and A. Garbali and R.A. Pimenta for trying the functional method for the nineteen-vertex model. For the revised version of this review, I further thank the anonymous referee for valuable feedback, A.G. Pronko for correspondence, and H. Rosengren for various useful discussions. The works [20, 31, 32] were supported by the Vici grant 680-47-602 and by the ERC Advanced Grant 246974, Supersymmetry: a window to non-perturbative physics, and part of the d-itp consortium, an NWO program funded by the Dutch Ministry of Education, Culture and Science (OCW). The present review was written with the support of the Knut and Alice Wallenberg Foundation (KAW). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Functional Method for the Domain-Wall Partition Function Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
The Functional Method for the Domain-Wall Partition Function |
| spellingShingle |
The Functional Method for the Domain-Wall Partition Function Lamers, J. |
| title_short |
The Functional Method for the Domain-Wall Partition Function |
| title_full |
The Functional Method for the Domain-Wall Partition Function |
| title_fullStr |
The Functional Method for the Domain-Wall Partition Function |
| title_full_unstemmed |
The Functional Method for the Domain-Wall Partition Function |
| title_sort |
functional method for the domain-wall partition function |
| author |
Lamers, J. |
| author_facet |
Lamers, J. |
| publishDate |
2018 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We review the (algebraic-)functional method devised by Galleas and further developed by Galleas and the author. We first explain the method using the simplest example: the computation of the partition function for the six-vertex model with domain-wall boundary conditions. At the heart of the method lies a linear functional equation for the partition function. After deriving this equation, we outline its analysis. The result is a closed expression in the form of a symmetrized sum - or, equivalently, multiple-integral formula - that can be rewritten to recover Izergin's determinant. Special attention is paid to the relation with other approaches. In particular, we show that the Korepin-Izergin approach can be recovered within the functional method. We comment on the functional method's range of applicability, and review how it is adapted to the technically more involved example of the elliptic solid-on-solid model with domain walls and a reflecting end. We present a new formula for the partition function of the latter, which was expressed as a determinant by Tsuchiya-Filali-Kitanine. Our result takes the form of a "crossing-symmetrized" sum with 2^L terms featuring the elliptic domain-wall partition function, which appears to be new also in the limiting case of the six-vertex model. Further taking the rational limit, we recover the expression obtained by Frassek using the boundary perimeter Bethe ansatz.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/209786 |
| citation_txt |
The Functional Method for the Domain-Wall Partition Function / J. Lamers // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 45 назв. — англ. |
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2025-12-07T18:53:20Z |
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2025-12-07T18:53:20Z |
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1850886105484754944 |