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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ftunivdauphine:oai:basepub.dauphine.psl.eu:123456789/23359 2024-09-09T19:26:07+00:00 The Schwarzian octahedron recurrence (dSKP equation) I : explicit solutions Affolter, Niklas de Tilière, Béatrice Melotti, Paul 2024-07-23T12:35:22Z application/pdf https://basepub.dauphine.psl.eu/handle/123456789/23359 en eng Paris Combinatorial Theory 3 2 2023 1-58 10.5070/C63261993 eScholarship oui 2766-1334 https://basepub.dauphine.psl.eu/handle/123456789/23359 Dimer model octahedron recurrence discrete KP equation integrable system spanning forests algebraic entropy discrete geometry projective geometry Aztec diamond limit shapes 515 Analyse Article accepté pour publication ou publié 2024 ftunivdauphine 2024-07-30T23:38:59Z 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]. non non recherche International Article in Journal/Newspaper Arctic Base Institutionnelle de Recherche de l'université Paris-Dauphine (BIRD) Arctic |
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Base Institutionnelle de Recherche de l'université Paris-Dauphine (BIRD) |
op_collection_id |
ftunivdauphine |
language |
English |
topic |
Dimer model octahedron recurrence discrete KP equation integrable system spanning forests algebraic entropy discrete geometry projective geometry Aztec diamond limit shapes 515 Analyse |
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Dimer model octahedron recurrence discrete KP equation integrable system spanning forests algebraic entropy discrete geometry projective geometry Aztec diamond limit shapes 515 Analyse Affolter, Niklas de Tilière, Béatrice Melotti, Paul The Schwarzian octahedron recurrence (dSKP equation) I : explicit solutions |
topic_facet |
Dimer model octahedron recurrence discrete KP equation integrable system spanning forests algebraic entropy discrete geometry projective geometry Aztec diamond limit shapes 515 Analyse |
description |
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]. non non recherche International |
format |
Article in Journal/Newspaper |
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 |
Paris |
publishDate |
2024 |
url |
https://basepub.dauphine.psl.eu/handle/123456789/23359 |
geographic |
Arctic |
geographic_facet |
Arctic |
genre |
Arctic |
genre_facet |
Arctic |
op_relation |
Combinatorial Theory 3 2 2023 1-58 10.5070/C63261993 eScholarship oui 2766-1334 https://basepub.dauphine.psl.eu/handle/123456789/23359 |
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1809895798213705728 |