The Schwarzian octahedron recurrence (dSKP equation) I: explicit solutions
We prove an explicit expression for the solutions of the discrete Schwarzian octahedron recurrence, also known as the discrete Schwarzian KP equation (dSKP), as the ratio of two partition functions. Each one counts weighted oriented dimer configurations of an associated bipartite graph, and is equal...
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ftanrparis:oai:HAL:hal-03765650v1 2024-09-09T19:25:48+00:00 The Schwarzian octahedron recurrence (dSKP equation) I: explicit solutions Affolter, Niklas de Tilière, Béatrice Melotti, Paul Institute of Mathematics, TU Berlin Technical University of Berlin / Technische Universität Berlin (TUB) CEntre de REcherches en MAthématiques de la DEcision (CEREMADE) Université Paris Dauphine-PSL Université Paris Sciences et Lettres (PSL)-Université Paris Sciences et Lettres (PSL)-Centre National de la Recherche Scientifique (CNRS) Laboratoire de Mathématiques d'Orsay (LMO) Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS) ANR-18-CE40-0033,DIMERS,Dimères : de la combinatoire à la mécanique quantique(2018) 2022-08-31 https://hal.science/hal-03765650 https://hal.science/hal-03765650/document https://hal.science/hal-03765650/file/2208.00239.pdf en eng HAL CCSD info:eu-repo/semantics/altIdentifier/arxiv/2208.00239 hal-03765650 https://hal.science/hal-03765650 https://hal.science/hal-03765650/document https://hal.science/hal-03765650/file/2208.00239.pdf ARXIV: 2208.00239 info:eu-repo/semantics/OpenAccess https://hal.science/hal-03765650 2022 [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] info:eu-repo/semantics/preprint Preprints, Working Papers, . 2022 ftanrparis 2024-07-12T10:58:28Z We prove an explicit expression for the solutions of the discrete Schwarzian octahedron recurrence, also known as the discrete Schwarzian KP equation (dSKP), as the ratio of two partition functions. Each one counts weighted oriented dimer configurations of an associated bipartite graph, and is equal to the determinant of a Kasteleyn matrix. This is in the spirit of Speyer's result on the dKP equation, or octahedron recurrence [Spe07]. One consequence is that dSKP has zero algebraic entropy, meaning that the growth of the degrees of the polynomials involved is only polynomial. There are cancellations in the partition function, and we prove an alternative, cancellation free explicit expression involving complementary trees and forests. Using all of the above, we show several instances of the Devron property for dSKP, i.e., that certain singularities in initial data repeat after a finite number of steps. This has many applications for discrete geometric systems and is the subject of the companion paper [AdTM22]. We also prove limit shape results analogous to the arctic circle of the Aztec diamond. Finally, we discuss the combinatorics of all the other octahedral equations in the classification of Adler, Bobenko and Suris [ABS12]. Report Arctic Portail HAL-ANR (Agence Nationale de la Recherche) Arctic |
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[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] Affolter, Niklas de Tilière, Béatrice Melotti, Paul The Schwarzian octahedron recurrence (dSKP equation) I: explicit solutions |
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[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] |
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We prove an explicit expression for the solutions of the discrete Schwarzian octahedron recurrence, also known as the discrete Schwarzian KP equation (dSKP), as the ratio of two partition functions. Each one counts weighted oriented dimer configurations of an associated bipartite graph, and is equal to the determinant of a Kasteleyn matrix. This is in the spirit of Speyer's result on the dKP equation, or octahedron recurrence [Spe07]. One consequence is that dSKP has zero algebraic entropy, meaning that the growth of the degrees of the polynomials involved is only polynomial. There are cancellations in the partition function, and we prove an alternative, cancellation free explicit expression involving complementary trees and forests. Using all of the above, we show several instances of the Devron property for dSKP, i.e., that certain singularities in initial data repeat after a finite number of steps. This has many applications for discrete geometric systems and is the subject of the companion paper [AdTM22]. We also prove limit shape results analogous to the arctic circle of the Aztec diamond. Finally, we discuss the combinatorics of all the other octahedral equations in the classification of Adler, Bobenko and Suris [ABS12]. |
author2 |
Institute of Mathematics, TU Berlin Technical University of Berlin / Technische Universität Berlin (TUB) CEntre de REcherches en MAthématiques de la DEcision (CEREMADE) Université Paris Dauphine-PSL Université Paris Sciences et Lettres (PSL)-Université Paris Sciences et Lettres (PSL)-Centre National de la Recherche Scientifique (CNRS) Laboratoire de Mathématiques d'Orsay (LMO) Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS) ANR-18-CE40-0033,DIMERS,Dimères : de la combinatoire à la mécanique quantique(2018) |
format |
Report |
author |
Affolter, Niklas de Tilière, Béatrice Melotti, Paul |
author_facet |
Affolter, Niklas de Tilière, Béatrice Melotti, Paul |
author_sort |
Affolter, Niklas |
title |
The Schwarzian octahedron recurrence (dSKP equation) I: explicit solutions |
title_short |
The Schwarzian octahedron recurrence (dSKP equation) I: explicit solutions |
title_full |
The Schwarzian octahedron recurrence (dSKP equation) I: explicit solutions |
title_fullStr |
The Schwarzian octahedron recurrence (dSKP equation) I: explicit solutions |
title_full_unstemmed |
The Schwarzian octahedron recurrence (dSKP equation) I: explicit solutions |
title_sort |
schwarzian octahedron recurrence (dskp equation) i: explicit solutions |
publisher |
HAL CCSD |
publishDate |
2022 |
url |
https://hal.science/hal-03765650 https://hal.science/hal-03765650/document https://hal.science/hal-03765650/file/2208.00239.pdf |
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Arctic |
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Arctic |
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https://hal.science/hal-03765650 2022 |
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info:eu-repo/semantics/altIdentifier/arxiv/2208.00239 hal-03765650 https://hal.science/hal-03765650 https://hal.science/hal-03765650/document https://hal.science/hal-03765650/file/2208.00239.pdf ARXIV: 2208.00239 |
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info:eu-repo/semantics/OpenAccess |
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1809895529551757312 |