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For a discrete time multitype supercritical Galton–Watson process (Z n) n ∈ N and corresponding genealogical tree T , we associate a new discrete time process (Z n Φ) n ∈ N such that, for each n ∈ N , the contribution of each individual u ∈ T to Z n Φ is determined by a (random) characteristic Φ evaluated at the age of u at time n. In other words, Z n Φ is obtained by summing over all u ∈ T the corresponding contributions Φ u , where (Φ u) u ∈ T are i.i.d. copies of Φ. Such processes are known in the literature under the name of Crump–Mode–Jagers (CMJ) processes counted with characteristic Φ. We derive a LLN and a CLT for the process (Z n Φ) n ∈ N in the discrete time setting, and in particular, we show a dichotomy in its limit behavior. By applying our main result, we also obtain a generalization of the results in Kesten and Stigum (1966). [ABSTRACT FROM AUTHOR]

The divisible sandpile model is a growth model on graphs that was introduced by Levine and Peres [Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile, Potential Anal. 30(1) (2009) 1–27] as a tool to study internal diffusion limited aggregation. In this work, we investigate the shape of the divisible sandpile model on the graphical Sierpinski gasket S G. We show that the shape is a ball in the graph metric of S G. Moreover, we give an exact representation of the odometer function of the divisible sandpile. [ABSTRACT FROM AUTHOR]

In this article, we investigate the connectedness and the isomorphism problems for zig-zag products of two graphs. A sufficient condition for the zig-zag product of two graphs to be connected is provided, reducing to the study of the connectedness property of a new graph which depends only on the second factor of the graph product. We show that, when the second factor is a cycle graph, the study of the isomorphism problem for the zig-zag product is equivalent to the study of the same problem for the associated pseudo-replacement graph. The latter is defined in a natural way, by a construction generalizing the classical replacement product, and its degree is smaller than the degree of the zig-zag product graph. Two particular classes of products are studied in detail: the zig-zag product of a complete graph with a cycle graph, and the zig-zag product of a 4-regular graph with the cycle graph of length 4. Furthermore, an example coming from the theory of Schreier graphs associated with the action of self-similar groups is also considered: the graph products are completely determined and their spectral analysis is developed. [ABSTRACT FROM AUTHOR]