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Finite Markov Chains and Algorithmic Applications
1. Basics of probability theory 2. Markov chains 3. Computer simulation of Markov chains 4. Irreducible and aperiodic Markov chains 5. Stationary distributions 6. Reversible Markov chains 7. MarkovExpand
Infinite clusters in dependent automorphism invariant percolation on trees
We study dependent bond percolation on the homogeneous tree T n of order n ≥ 2 under the assumption of automorphism invariance. Excluding a trivial case, we find that the number of infinite clustersExpand
Nearest neighbor and hard sphere models in continuum percolation
It is shown that if percolation occurs, then there is exactly one infinite cluster and some general properties of percolations models where balls placed at Poisson points are not allowed to overlap (but are allowed to be tangent). Expand
Random-cluster measures and uniform spanning trees
Consider the random-cluster model on the integer lattice with parameters p and q. As p, q --> 0 in such a way that q/p --> 0, the random-cluster measures converge weakly to the uniform spanning treeExpand
Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously
Abstract. Consider site or bond percolation with retention parameter p on an infinite Cayley graph. In response to questions raised by Grimmett and Newman (1990) and Benjamini and Schramm (1996), weExpand
Percolation transitive graphs as a coalescent process: relentless merging followed by simultaneous uniqueness
Consider i.i.d. percolation with retention parameter p on an infinite graph G. There is a well known critical parameter p c ∈ [0, 1] for the existence of infinite open clusters. Recently, it has beenExpand
Absence of mutual unbounded growth for almost all parameter values in the two-type Richardson model
We study the two-type Richardson model on , d[greater-or-equal, slanted]2, in the asymmetric case where the two particle types have different infection rates. Starting with a single particle of eachExpand
Phase transition in continuum Potts models
We establish phase transitions for a class of continuum multi-type particle systems with finite range repulsive pair interaction between particles of different type. This proves an old conjecture ofExpand
Coloring percolation clusters at random
We consider the random coloring of the vertices of a graph G, that arises by first performing i.i.d. bond percolation with parameter p on G, and then assigning a random color, chosen according toExpand
Instability of matchings in decentralized markets with various preference structures
The main result is that even the outcome of decentralized matching with incomplete information can be expected to be “almost stable” under reasonable assumptions. Expand