714 research outputs found
Natural spanning trees of Zd are recurrent
AbstractWe show that the simple random walk on the natural spanning of Zd is recurrent for every d(= 1, 2, 3, …) and determine the asymptotic behaviour of the probability of returning to the origin in n steps (n → ∞). This is in contrast to a result of Polya [6]: Zd is recurrent for d = 1, 2 and transient for d⩾3
Infinite volume limit of the Abelian sandpile model in dimensions d >= 3
We study the Abelian sandpile model on Z^d. In dimensions at least 3 we prove
existence of the infinite volume addition operator, almost surely with respect
to the infinite volume limit mu of the uniform measures on recurrent
configurations. We prove the existence of a Markov process with stationary
measure mu, and study ergodic properties of this process. The main techniques
we use are a connection between the statistics of waves and uniform
two-component spanning trees and results on the uniform spanning tree measure
on Z^d.Comment: First version: LaTeX; 29 pages. Revised version: LaTeX; 29 pages. The
main result of the paper has been extended to all dimensions at least 3, with
a new and simplyfied proof of finiteness of the two-component spanning tree.
Second revision: LaTeX; 32 page
Anchored burning bijections on finite and infinite graphs
Let be an infinite graph such that each tree in the wired uniform
spanning forest on has one end almost surely. On such graphs , we give a
family of continuous, measure preserving, almost one-to-one mappings from the
wired spanning forest on to recurrent sandpiles on , that we call
anchored burning bijections. In the special case of , ,
we show how the anchored bijection, combined with Wilson's stacks of arrows
construction, as well as other known results on spanning trees, yields a power
law upper bound on the rate of convergence to the sandpile measure along any
exhaustion of . We discuss some open problems related to these
findings.Comment: 26 pages; 1 EPS figure. Minor alterations made after comments from
refere
The looping rate and sandpile density of planar graphs
We give a simple formula for the looping rate of loop-erased random walk on a
finite planar graph. The looping rate is closely related to the expected amount
of sand in a recurrent sandpile on the graph. The looping rate formula is
well-suited to taking limits where the graph tends to an infinite lattice, and
we use it to give an elementary derivation of the (previously computed) looping
rate and sandpile densities of the square, triangular, and honeycomb lattices,
and compute (for the first time) the looping rate and sandpile densities of
many other lattices, such as the kagome lattice, the dice lattice, and the
truncated hexagonal lattice (for which the values are all rational), and the
square-octagon lattice (for which it is transcendental)
Sandpile models
This survey is an extended version of lectures given at the Cornell
Probability Summer School 2013. The fundamental facts about the Abelian
sandpile model on a finite graph and its connections to related models are
presented. We discuss exactly computable results via Majumdar and Dhar's
method. The main ideas of Priezzhev's computation of the height probabilities
in 2D are also presented, including explicit error estimates involved in
passing to the limit of the infinite lattice. We also discuss various questions
arising on infinite graphs, such as convergence to a sandpile measure, and
stabilizability of infinite configurations.Comment: 72 pages - v3 incorporates referee's comments. References closely
related to the lectures were added/update
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
Minimal spanning forests
Minimal spanning forests on infinite graphs are weak limits of minimal
spanning trees from finite subgraphs. These limits can be taken with free or
wired boundary conditions and are denoted FMSF (free minimal spanning forest)
and WMSF (wired minimal spanning forest), respectively. The WMSF is also the
union of the trees that arise from invasion percolation started at all
vertices. We show that on any Cayley graph where critical percolation has no
infinite clusters, all the component trees in the WMSF have one end a.s. In
this was proved by Alexander [Ann. Probab. 23 (1995) 87--104],
but a different method is needed for the nonamenable case. We also prove that
the WMSF components are ``thin'' in a different sense, namely, on any graph,
each component tree in the WMSF has a.s., where
denotes the critical probability for having an infinite
cluster in Bernoulli percolation. On the other hand, the FMSF is shown to be
``thick'': on any connected graph, the union of the FMSF and independent
Bernoulli percolation (with arbitrarily small parameter) is a.s. connected. In
conjunction with a recent result of Gaboriau, this implies that in any Cayley
graph, the expected degree of the FMSF is at least the expected degree of the
FSF (the weak limit of uniform spanning trees). We also show that the number of
infinite clusters for Bernoulli() percolation is at most the
number of components of the FMSF, where denotes the critical
probability for having a unique infinite cluster. Finally, an example is given
to show that the minimal spanning tree measure does not have negative
associations.Comment: Published at http://dx.doi.org/10.1214/009117906000000269 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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