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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ftcdlib:oai:escholarship.org:ark:/13030/qt2jq67049 2023-10-09T21:49:15+02:00 The Schwarzian octahedron recurrence (dSKP equation) I: explicit solutions Affolter, Niklas Christoph de Tilière, Béatrice Melotti, Paul 2023-01-01 application/pdf https://escholarship.org/uc/item/2jq67049 unknown eScholarship, University of California qt2jq67049 https://escholarship.org/uc/item/2jq67049 CC-BY Combinatorial Theory, vol 3, iss 2 Dimer model octahedron recurrence discrete KP equation integrable system spanning forests algebraic entropy discrete geometry projective geometry Aztec diamond limit shapes article 2023 ftcdlib 2023-09-18T18:02:53Z 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 (Journal of Alg. Comb. 2007). 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 a companion paper (preprint 2022, Affolter, de Tillière, and Melotti). We also find 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 (IMRN 2012).Mathematics Subject Classifications: 05A15, 37K10, 37K60, 82B20, 82B23Keywords: Dimer model, octahedron recurrence, discrete KP equation, integrable system, spanning forests, algebraic entropy, discrete geometry, projective geometry, Aztec diamond, limit shapes Article in Journal/Newspaper Arctic University of California: eScholarship Arctic |
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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 |
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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 Affolter, Niklas Christoph 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 |
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 (Journal of Alg. Comb. 2007). 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 a companion paper (preprint 2022, Affolter, de Tillière, and Melotti). We also find 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 (IMRN 2012).Mathematics Subject Classifications: 05A15, 37K10, 37K60, 82B20, 82B23Keywords: Dimer model, octahedron recurrence, discrete KP equation, integrable system, spanning forests, algebraic entropy, discrete geometry, projective geometry, Aztec diamond, limit shapes |
format |
Article in Journal/Newspaper |
author |
Affolter, Niklas Christoph de Tilière, Béatrice Melotti, Paul |
author_facet |
Affolter, Niklas Christoph de Tilière, Béatrice Melotti, Paul |
author_sort |
Affolter, Niklas Christoph |
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 |
eScholarship, University of California |
publishDate |
2023 |
url |
https://escholarship.org/uc/item/2jq67049 |
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Arctic |
geographic_facet |
Arctic |
genre |
Arctic |
genre_facet |
Arctic |
op_source |
Combinatorial Theory, vol 3, iss 2 |
op_relation |
qt2jq67049 https://escholarship.org/uc/item/2jq67049 |
op_rights |
CC-BY |
_version_ |
1779312261645467648 |