The Cube Recurrence

Carroll, Gabriel; Speyer, David E

doi:10.37236/1826

Cited by 38 publications

(47 citation statements)

References 8 publications

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“…The edge-variables version of the cube recurrence gives g 0,0,0 as a rational function in the variables {a j,k , b i,k , c i,j , g i,j,k } (i,j,k)∈I . The following is the main result of [CS04].…”

Section: Groves and The Cube Recurrencementioning

confidence: 87%

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Grove Arctic Curves from Periodic Cluster Modular Transformations

George

2020

International Mathematics Research Notices

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Groves are spanning forests of a finite region of the triangular lattice that are in bijection with Laurent monomials that arise in solutions of the cube recurrence. We introduce a large class of probability measures on groves for which we can compute exact generating functions for edge probabilities. Using the machinery of asymptotics of multivariate generating functions, this lets us explicitly compute arctic curves, generalizing the arctic circle theorem of Petersen and Speyer. Our class of probability measures is sufficiently general that the limit shapes exhibit all solid and gaseous phases expected from the classification of EGMs in the resistor network model.

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Section: Groves and The Cube Recurrencementioning

confidence: 87%

“…It is shown in [CS04] that groves on standard initial conditions I(n) are completely determined by their long diagonal edges. Therefore, we can represent groves as a spanning forest of a finite portion of the triangular lattice(see Figure 1), which is called a simplified grove.…”

Section: Groves and The Cube Recurrencementioning

confidence: 99%

Grove Arctic Curves from Periodic Cluster Modular Transformations

George

2020

International Mathematics Research Notices

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“…The following lemmas are direct adaptations of Carroll and Speyer's arguments on cube groves [5]. Proof.…”

Section: Boundary Conditionsmentioning

confidence: 90%

“…We compute some limit shapes of the model by a now standard technique [33,11,29,19]. We also show that in characteristic 2 the model reduces to the cube groves of [5].…”

Section: Theorem For Any Free-fermionic Cmentioning

confidence: 99%

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The free-fermionic C2(1) loop model, double dimers and Kashaev's recurrence

Melotti

2018

Journal of Combinatorial Theory, Series A

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We study a two-color loop model known as the C (1) 2 loop model. We define a free-fermionic regime for this model, and show that under this assumption it can be transformed into a double dimer model. We then compute its free energy on periodic planar graphs. We also study the star-triangle relation or Yang-Baxter equations of this model, and show that after a proper parametrization they can be summed up into a single relation known as Kashaev's relation. This is enough to identify the solution of Kashaev's relation as the partition function of a C (1) 2 loop model with some boundary conditions, thus solving an open question of Kenyon and Pemantle [29] about the combinatorics of Kashaev's relation. 1 Kashaev's initial equation contained a +4 instead of a −4 coefficient for the last terms, but one can easily get from one to another, for instance by multiplying g by −1 at a vertex of the cube and its three neighbors.

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R-systems

Galashin

Pylyavskyy

2019

Sel. Math. New Ser.

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Birational toggling on Gelfand-Tsetlin patterns appeared first in the study of geometric crystals and geometric Robinson-Schensted-Knuth correspondence. Based on these birational toggle relations, Einstein and Propp introduced a discrete dynamical system called birational rowmotion associated with a partially ordered set. We generalize birational rowmotion to the class of arbitrary strongly connected directed graphs, calling the resulting discrete dynamical system the R-system. We study its integrability from the points of view of singularity confinement and algebraic entropy. We show that in many cases, singularity confinement in an R-system reduces to the Laurent phenomenon either in a cluster algebra, or in a Laurent phenomenon algebra, or beyond both of those generalities, giving rise to many new sequences with the Laurent property possessing rich groups of symmetries. Some special cases of R-systems reduce to Somos and Gale-Robinson sequences.

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The Cube Recurrence

Cited by 38 publications

References 8 publications

Grove Arctic Curves from Periodic Cluster Modular Transformations

Grove Arctic Curves from Periodic Cluster Modular Transformations

The free-fermionic C2(1) loop model, double dimers and Kashaev's recurrence

R-systems

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