43 research outputs found
The Cube Recurrence
We construct a combinatorial model that is described by the cube recurrence,
a nonlinear recurrence relation introduced by Propp, which generates families
of Laurent polynomials indexed by points in . In the process, we
prove several conjectures of Propp and of Fomin and Zelevinsky, and we obtain a
combinatorial interpretation for the terms of Gale-Robinson sequences. We also
indicate how the model might be used to obtain some interesting results about
perfect matchings of certain bipartite planar graphs
The Laurent phenomenon
A composition of birational maps given by Laurent polynomials need not be
given by Laurent polynomials; however, sometimes---quite unexpectedly---it
does. We suggest a unified treatment of this phenomenon, which covers a large
class of applications. In particular, we settle in the affirmative a conjecture
of D.Gale and R.Robinson on integrality of generalized Somos sequences, and
prove the Laurent property for several multidimensional recurrences, confirming
conjectures by J.Propp, N.Elkies, and M.Kleber.Comment: 21 page
A Periodicity Theorem for the Octahedron Recurrence
We investigate a variant of the octahedron recurrence which lives in a
3-dimensional lattice contained in [0,n] x [0,m] x R. Generalizing results of
David Speyer math.CO/0402452, we give an explicit non-recursive formula for the
values of this recurrence in terms of perfect matchings. We then use it to
prove that the octahedron recurrence is periodic of period n+m. This result is
reminiscent of Fomin and Zelevinsky's theorem about the periodicity of
Y-systems.Comment: 22 pages, (a few pictures added, section 3 has been reorganized
Dimers and cluster integrable systems
We show that the dimer model on a bipartite graph on a torus gives rise to a
quantum integrable system of special type - a cluster integrable system. The
phase space of the classical system contains, as an open dense subset, the
moduli space of line bundles with connections on the graph. The sum of
Hamiltonians is essentially the partition function of the dimer model. Any
graph on a torus gives rise to a bipartite graph on the torus. We show that the
phase space of the latter has a Lagrangian subvariety. We identify it with the
space parametrizing resistor networks on the original graph.We construct
several discrete quantum integrable systems.Comment: This is an updated version, 75 pages, which will appear in Ann. Sci.
EN
The B-quadrilateral lattice, its transformations and the algebro-geometric construction
The B-quadrilateral lattice (BQL) provides geometric interpretation of Miwa's
discrete BKP equation within the quadrialteral lattice (QL) theory. After
discussing the projective-geometric properties of the lattice we give the
algebro-geometric construction of the BQL ephasizing the role of Prym varieties
and the corresponding theta functions. We also present the reduction of the
vectorial fundamental transformation of the QL to the BQL case.Comment: 23 pages, 3 figures; presentation improved, some typos correcte
The electrical response matrix of a regular 2n-gon
Consider a unit-resistive plate in the shape of a regular polygon with 2n
sides, in which even-numbered sides are wired to electrodes and odd-numbered
sides are insulated. The response matrix, or Dirichlet-to-Neumann map, allows
one to compute the currents flowing through the electrodes when they are held
at specified voltages. We show that the entries of the response matrix of the
regular 2n-gon are given by the differences of cotangents of evenly spaced
angles, and we describe some connections with the limiting distributions of
certain random spanning forests.Comment: 10 pages, 4 figures; v2 adds more background informatio