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Summary. We study the following growth model on a regular d-ary tree. Points at distance n adjacent to the existing subtree are added with probabilities proportional to α[sup -n] , where α < 1 is a positive real parameter. The heights of these clusters are shown to increase linearly with their total size; this complements known results that show the height increases only logarithmically when α ≧ 1. Results are obtained using stochastic monotonicity and regeneration results which may be of independent interest. Our motivation comes from two other ways in which the model may be viewed: as a problem in first-passage percolation, and as a version of diffusion-limited aggregation (DLA), adjusted so that “fingering” occurs. [ABSTRACT FROM AUTHOR]

Summary. We show that with probability 1, the trace B[0, 1] of Brownian motion in space, has positive capacity with respect to exactly the same kernels as the unit square. More precisely, the energy of occupation measure on B[0, 1] in the kernel f(∣x-y∣), is bounded above and below by constant multiples of the energy of Lebesgue measure on the unit square. (The constants are random, but do not depend on the kernel.) As an application, we give almost-sure asymptotics for the probability that an α-stable process approaches within ε of B[0, 1], conditional on B[0, 1]. The upper bound on energy is based on a strong law for the approximate self-intersections of the Brownian path. We also prove analogous capacity estimates for planar Brownian motion and for the zero-set of one-dimensional Brownian motion. [ABSTRACT FROM AUTHOR]

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 transient. The corresponding harmonic measures are carried by subsets of the boundary of dimension smaller than that of the whole boundary. When the bias is directed away from the root and the extinction probability is positive, the speed may be zero even though the walk is transient; the critical bias for positive speed is determined. [ABSTRACT FROM AUTHOR]

This 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 each state has been visited and by a fixed a priori likelihood matrix, R, which is real, symmetric and nonnegative. Let S (n) keep track of the number of visits to state i up to time n, and form the fractional occupation vector, V( n), where $$v_i (n) = {{S_i (n)} \mathord{\left/ {\vphantom {{S_i (n)} {\left( {\sum\limits_{j = 1}^d {S_j (n)} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\sum\limits_{j = 1}^d {S_j (n)} } \right)}}$$ . It is shown that V( n) converges to to a set of critical points for the quadratic form H with matrix R, and that under nondegeneracy conditions on R, there is a finite set of points such that with probability one, V( n)→ p for some p in the set. There may be more than one p in this set for which P(V( n)→ p)>0. On the other hand P(V( n)→ p)=0 whenever p fails in a strong enough sense to be maximum for H. [ABSTRACT FROM AUTHOR]