The models surveyed include generalized P\'{o}lya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of some proofs. Applications are discussed in statistics, biology, economics and a number of other areas.Comment: Published at http://dx.doi.org/10.1214/07-PS094 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

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, results of Kolmogorov and later authors give the rate of decay of the probability that the process survives at least n generations. We give conceptual proofs of these theorems based on comparisons of Galton-Watson measure to another measure on the space of trees. This approach also explains Yaglom's exponential limit law for conditioned critical branching processes via a simple characterization of the exponential distribution. §1. Introduction.Consider a Galton-Watson branching process with each particle having probability p k of generating k children. Let L stand for a random variable with this offspring distribution. Let m := k kp k be the mean number of children per particle and let Z n be the number of particles in the n th generation. The most basic and well-known fact about branching processes is that the extinction probability q := lim P[Z n = 0] is equal to 1 if and only if m ≤ 1 and p 1 < 1. It is also not hard to establish that in the case m > 1, 1 n log Z n → log m almost surely on nonextinction, while in the case m ≤ 1, 1 n log P[Z n > 0] → log m.Finer questions may be asked:• In the case m > 1, when does the mean E[Z n ] = m n give the right growth rate up to a random factor?• In the case m < 1, when does the first moment estimate P[Z n > 0] ≤ E[Z n ] = m n give the right decay rate up to a random factor?• In the case m = 1, what is the decay rate of P[Z n > 0]?These questions are answered by the following three classical theorems.1991 Mathematics Subject Classification. Primary 60J80.

Let G be a finite graph or an infinite graph on which Z 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 infinite, methods from [Pem] show that this still makes sense, producing a random essential spanning forest of G. A method for calculating local characteristics (i.e. finite-dimensional marginals) of T from the transfer-impedance matrix is presented. This differs from the classical matrix-tree theorem in that only small pieces of the matrix (n-dimensional minors) are needed to compute small (n-dimensional) marginals. Calculation of the matrix entries relies on the calculation of the Green's function for G, which is not a local calculation. However, it is shown how the calculation of the Green's function may be reduced to a finite computation in the case when G is an infinite graph admitting a Z d -action with finite quotient. The same computation also gives the entropy of the law of T.These results are applied to the problem of tiling certain lattices by dominos -the so-called dimer problem. Another application of these results is to prove modified versions of conjectures of Aldous [Al2] on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity. Included is a generalization of moments to tree-valued random variables and criteria for these generalized moments to determine a distribution.

Consider the nearest neighbor graph for the integer lattice Z d in d dimensions. For a large finite piece of it, consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees. As the piece gets larger, this approaches a limiting measure on the set of spanning graphs for Z d . This is shown to be a tree if and only if d ≤ 4. In this case, the tree has only one topological end, i.e. there are no doubly infinite paths. When d ≥ 5 the spanning forest has infinitely many components almost surely, with each component having one or two topological ends.

A molecular theory for the rank dependence of orientational relaxation in Brownian dipolar lattice P͑X 1 ϭ0,X 2 ϭ0,X 3 ϭ0 ͒ϭ16, 1379 P͑X 1 ϭ0,X 2 ϭ0,X 3 ϭ1 ͒ϭ8, P͑X 1 ϭ0,X 2 ϭ1,X 3 ϭ0 ͒ϭ8, P͑X 1 ϭ0,X 2 ϭ1,X 3 ϭ1 ͒ϭ8, P͑X 1 ϭ1,X 2 ϭ0,X 3 ϭ0 ͒ϭ12ϩ⑀,When 0р⑀р0.8 then this measure satisfies CNA and hence JNRD and h-NLC. However, when ⑀Ͼ0, then applying the external field ͑, 1, 1͒ for any positive Ͻ⑀/(1Ϫ⑀) yields a measure in which X 2 and X 3 are positively correlated, thus violating h-NLC and hence JNRD and CNA. This shows the first three vertical implications in Diag. 1 are strict.Example 2: Suppose nϭ3, and the probabilities for the various possible atoms are in the proportions P͑X 1 ϭ0,X 2 ϭ0,X 3 ϭ0 ͒ϭ0, P͑X 1 ϭ0,X 2 ϭ0,X 3 ϭ1 ͒ϭ1, P͑X 1 ϭ0,X 2 ϭ1,X 3 ϭ0 ͒ϭ1, P͑X 1 ϭ0,X 2 ϭ1,X 3 ϭ1 ͒ϭ10⑀, P͑X 1 ϭ1,X 2 ϭ0,X 3 ϭ0 ͒ϭ1, P͑X 1 ϭ1,X 2 ϭ0,X 3 ϭ1 ͒ϭ1, P͑X 1 ϭ1,X 2 ϭ1,X 3 ϭ0 ͒ϭ10⑀, P͑X 1 ϭ1,X 2 ϭ1,X 3 ϭ1 ͒ϭ⑀.