5,329 research outputs found
Abelian sandpiles: an overview and results on certain transitive graphs
We review the Majumdar-Dhar bijection between recurrent states of the Abelian
sandpile model and spanning trees. We generalize earlier results of Athreya and
Jarai on the infinite volume limit of the stationary distribution of the
sandpile model on Z^d, d >= 2, to a large class of graphs. This includes: (i)
graphs on which the wired spanning forest is connected and has one end; (ii)
transitive graphs with volume growth at least c n^5 on which all bounded
harmonic functions are constant. We also extend a result of Maes, Redig and
Saada on the stationary distribution of sandpiles on infinite regular trees, to
arbitrary exhaustions.Comment: 44 pages. Version 2 incorporates some smaller changes. To appear in
Markov Processes and Related Fields in the proceedings of the meeting:
Inhomogeneous Random Systems, Stochastic Geometry and Statistical Mechanics,
Institut Henri Poincare, Paris, 27 January 201
Perturbation of transportation polytopes
We describe a perturbation method that can be used to reduce the problem of
finding the multivariate generating function (MGF) of a non-simple polytope to
computing the MGF of simple polytopes. We then construct a perturbation that
works for any transportation polytope. We apply this perturbation to the family
of central transportation polytopes of order kn x n, and obtain formulas for
the MGFs of the feasible cone of each vertex of the polytope and the MGF of the
polytope. The formulas we obtain are enumerated by combinatorial objects. A
special case of the formulas recovers the results on Birkhoff polytopes given
by the author and De Loera and Yoshida. We also recover the formula for the
number of maximum vertices of transportation polytopes of order kn x n.Comment: 25 pages, 3 figures. To appear in Journal of Combinatorial Theory
Ser.
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