1,564 research outputs found
Numerical Study of the Correspondence Between the Dissipative and Fixed Energy Abelian Sandpile Models
We consider the Abelian sandpile model (ASM) on the large square lattice with
a single dissipative site (sink). Particles are added by one per unit time at
random sites and the resulting density of particles is calculated as a function
of time. We observe different scenarios of evolution depending on the value of
initial uniform density (height) . During the first stage of the
evolution, the density of particles increases linearly. Reaching a critical
density , the system changes its behavior sharply and relaxes
exponentially to the stationary state of the ASM with . We found
numerically that and . Our
observations suggest that the equality holds for more general
initial conditions with non-positive heights. In parallel with the ASM, we
consider the conservative fixed-energy Abelian sandpile model (FES). The
extensive Monte-Carlo simulations for have confirmed that in the
limit of large lattices coincides with the threshold density
of FES. Therefore, can be identified with
if the FES starts its evolution with non-positive uniform height .Comment: 6 pages, 8 figure
The problem of predecessors on spanning trees
We consider the equiprobable distribution of spanning trees on the square
lattice. All bonds of each tree can be oriented uniquely with respect to an
arbitrary chosen site called the root. The problem of predecessors is finding
the probability that a path along the oriented bonds passes sequentially fixed
sites and . The conformal field theory for the Potts model predicts the
fractal dimension of the path to be 5/4. Using this result, we show that the
probability in the predecessors problem for two sites separated by large
distance decreases as . If sites and are
nearest neighbors on the square lattice, the probability can be
found from the analytical theory developed for the sandpile model. The known
equivalence between the loop erased random walk (LERW) and the directed path on
the spanning tree says that is the probability for the LERW started at
to reach the neighboring site . By analogy with the self-avoiding walk,
can be called the return probability. Extensive Monte-Carlo simulations
confirm the theoretical predictions.Comment: 7 pages, 2 figure
Jamming probabilities for a vacancy in the dimer model
Following the recent proposal made by Bouttier et al [Phys. Rev. E 76, 041140
(2007)], we study analytically the mobility properties of a single vacancy in
the close-packed dimer model on the square lattice. Using the spanning web
representation, we find determinantal expressions for various observable
quantities. In the limiting case of large lattices, they can be reduced to the
calculation of Toeplitz determinants and minors thereof. The probability for
the vacancy to be strictly jammed and other diffusion characteristics are
computed exactly.Comment: 19 pages, 6 figure
Transfer matrix for spanning trees, webs and colored forests
We use the transfer matrix formalism for dimers proposed by Lieb, and
generalize it to address the corresponding problem for arrow configurations (or
trees) associated to dimer configurations through Temperley's correspondence.
On a cylinder, the arrow configurations can be partitioned into sectors
according to the number of non-contractible loops they contain. We show how
Lieb's transfer matrix can be adapted in order to disentangle the various
sectors and to compute the corresponding partition functions. In order to
address the issue of Jordan cells, we introduce a new, extended transfer
matrix, which not only keeps track of the positions of the dimers, but also
propagates colors along the branches of the associated trees. We argue that
this new matrix contains Jordan cells.Comment: 29 pages, 7 figure
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