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Choosing a Spanning Tree for the Integer Lattice Uniformly
Consider the nearest neighbor graph for the integer lattice Zd in d dimensions. For a large finite piece of it, consider choosing a spanning tree for that piece uniformly among all possible subgraphs
Conceptual proofs of L log L criteria for mean behavior of branching processes
The Kesten-Stigum theorem is a fundamental criterion for the rate of growth of a supercritical branching process, showing that an L log L condition is decisive. In critical and subcritical cases,
A survey of random processes with reinforcement
The models surveyed include generalized Polya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided
Local Characteristics, Entropy and Limit Theorems for Spanning Trees and Domino Tilings Via Transfer-Impedances
Let G be a finite graph or an infinite graph on which Z^d acts with finite fundamental domain. If G is finite, let T be a random spanning tree chosen uniformly from all spanning trees of G; if G is
Nonconvergence to Unstable Points in Urn Models and Stochastic Approximations
A particle in Rd moves in discrete time. The size of the nth step is of order 1/n and when the particle is at a position v the expectation of the next step is in the direction F(v) for some fixed
Random walk in a random environment and rst-passage percolation on trees
A delay line refresh memory stores the bits to be displayed on a visual display means such as a television receiver. A shift register in the feedback loop applies the stored bits back to the input
Towards a theory of negative dependence
The FKG theorem says that the positive lattice condition, an easily checkable hypothesis which holds for many natural families of events, implies positive association, a very useful property. Thus
Vertex-reinforced random walk
SummaryThis paper considers a class of non-Markovian discrete-time random processes on a finite state space {1,...,d}. The transition probabilities at each time are influenced by the number of times
Ergodic theory on Galton—Watson trees: speed of random walk and dimension of harmonic measure
Abstract We consider simple random walk on the family tree T of a nondegenerate supercritical Galton—Watson branching process and show that the resulting harmonic measure has a.s. strictly smaller
Biased random walks on Galton–Watson trees
Summary. We consider random walks with a bias toward the root on the family tree T of a supercritical Galton–Watson branching process and show that the speed is positive whenever the walk is