119 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
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
Phase transition in a sequential assignment problem on graphs
We study the following game on a finite graph . At the start,
each edge is assigned an integer , . In
round , , a uniformly random vertex is chosen and
one of the edges incident with is selected by the player. The value
assigned to is then decreased by . The player wins, if the configuration
is reached; in other words, the edge values never go negative.
Our main result is that there is a phase transition: as , the
probability that the player wins approaches a constant when converges to a point in the interior of a certain convex set
, and goes to exponentially when is
bounded away from . We also obtain upper bounds in the
near-critical region, that is when lies close to . We supply quantitative error bounds in our arguments.Comment: 28 pages, 2 eps figures. Some mistakes have been corrected, and the
introduction has been re-written. Minor corrections throughou
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
Electrical resistance of the low dimensional critical branching random walk
We show that the electrical resistance between the origin and generation n of
the incipient infinite oriented branching random walk in dimensions d<6 is
O(n^{1-alpha}) for some universal constant alpha>0. This answers a question of
Barlow, J\'arai, Kumagai and Slade [2].Comment: 44 pages, 3 figure
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