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Let $T$ be an infinite rooted tree with weights $w_e$ assigned to its edges. Denote by $m_n(T)$ the minimum weight of a path from the root to a node of the $n$th generation. We consider the possible behaviour of $m_n(T)$ with focus on the two following cases: we say $T$ is explosive if [ lim_{nto infty}m_n(T) < infty, ] and say that $T$ exhibits linear growth if [ liminf_{nto infty} frac{m_n(T)}{n} > 0. ] We consider a class of infinite randomly weighted trees related to the Poisson-weighted infinite tree, and determine precisely which trees in this class have linear growth almost surely. We then apply this characterization to obtain new results concerning the event of explosion in infinite randomly weighted spherically-symmetric trees, answering a question of Pemantle and Peres. As a further application, we consider the random real tree generated by attaching sticks of deterministic decreasing lengths, and determine for which sequences of lengths the tree has finite height almost surely.
Consider a uniformly sampled random $d$-regular graph on $n$ vertices. If $d$ is fixed and $n$ goes to $infty$ then we can relate typical (large probability) properties of such random graph to a family of invariant random processes (called typical pr
Consider a statistical physical model on the $d$-regular infinite tree $T_{d}$ described by a set of interactions $Phi$. Let ${G_{n}}$ be a sequence of finite graphs with vertex sets $V_n$ that locally converge to $T_{d}$. From $Phi$ one can construc
In order to approximate the exit time of a one-dimensional diffusion process, we propose an algorithm based on a random walk. Such an algorithm was already introduced in both the Brownian context and in the Ornstein-Uhlenbeck context. Here the aim is
We introduce the notion of linear multifractional stable sheets in the broad sense (LMSS) with $alphain(0,2]$, to include both linear multifractional Brownian sheets ($alpha=2$) and linear multifractional stable sheets ($alpha<2$). The purpose of the
We study some percolation problems on the complete graph over $mathbf N$. In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the probability,