Extendable self-avoiding walks

The connective constant mu of a graph is the exponential growth rate of the number of n-step self-avoiding walks starting at a given vertex. A self-avoiding walk is said to be forward (respectively, backward) extendable if it may be extended forwards (respectively, backwards) to a singly infinite self-avoiding walk. It is called doubly extendable if it may be extended in both directions simultaneously to a doubly infinite self-avoiding walk. We prove that the connective constants for forward, backward, and doubly extendable self-avoiding walks, denoted respectively by mu^F, mu^B, mu^FB, exist and satisfy mu = mu^F = mu^B = mu^FB for every infinite, locally finite, strongly connected, quasi-transitive directed graph. The proofs rely on a 1967 result of Furstenberg on dimension, and involve two different arguments depending on whether or not the graph is unimodular.Comment: Accepted versio

Conjugate Projective Limits

We characterize conjugate nonparametric Bayesian models as projective limits of conjugate, finite-dimensional Bayesian models. In particular, we identify a large class of nonparametric models representable as infinite-dimensional analogues of exponential family distributions and their canonical conjugate priors. This class contains most models studied in the literature, including Dirichlet processes and Gaussian process regression models. To derive these results, we introduce a representation of infinite-dimensional Bayesian models by projective limits of regular conditional probabilities. We show under which conditions the nonparametric model itself, its sufficient statistics, and -- if they exist -- conjugate updates of the posterior are projective limits of their respective finite-dimensional counterparts. We illustrate our results both by application to existing nonparametric models and by construction of a model on infinite permutations.Comment: 49 pages; improved version: revised proof of theorem 3 (results unchanged), discussion added, exposition revise

Uniqueness and multiplicity of infinite clusters

The Burton--Keane theorem for the almost-sure uniqueness of infinite clusters is a landmark of stochastic geometry. Let $\mu$ be a translation-invariant probability measure with the finite-energy property on the edge-set of a $d$-dimensional lattice. The theorem states that the number $I$ of infinite components satisfies $\mu(I\in\{0,1\})=1$. The proof is an elegant and minimalist combination of zero--one arguments in the presence of amenability. The method may be extended (not without difficulty) to other problems including rigidity and entanglement percolation, as well as to the Gibbs theory of random-cluster measures, and to the central limit theorem for random walks in random reflecting labyrinths. It is a key assumption on the underlying graph that the boundary/volume ratio tends to zero for large boxes, and the picture for non-amenable graphs is quite different.Comment: Published at http://dx.doi.org/10.1214/074921706000000040 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

On cardinalities in quotients of inverse limits of groups

Let lambda be aleph_0 or a strong limit of cofinality aleph_0. Suppose that (G_m,p_{m,n}:m =< n<omega) and (H_m,p^t_{m,n}: m=< n < omega) are projective systems of groups of cardinality less than lambda and suppose that for every nG_n such that all the diagrams commute. If for every mu<lambda there exists (f_i in G_omega:i<mu) such that for distinct i,j we have: f_i f_j^{-1} notin h_omega(H_omega), then there exists (f_i in G_omega:i<2^lambda) such that for distinct i,j we have f_i f_j^{-1} notin h_omega(H_omega)