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Register a new userRumor processes in random environment on N and on Galton-Watson trees
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The aim of this paper is to study rumor processes in random environment. In a rumor process a signal starts from the stations of a fixed vertex (the root) and travels on a graph from vertex to vertex. We consider two rumor processes. In the firework process each station, when reached by the signal, transmits it up to a random distance. In the reverse firework process, on the other hand, stations do not send any signal but they listen for it up to a random distance. The first random environment that we consider is the deterministic 1-dimensional tree N with a random number of stations on each vertex; in this case the root is the origin of N. We give conditions for the survival/extinction on almost every realization of the sequence of stations. Later on, we study the processes on Galton-Watson trees with random number of stations on each vertex. We show that if the probability of survival is positive, then there is survival on almost every realization of the infinite tree such that there is at least one station at the root. We characterize the survival of the process in some cases and we give sufficient conditions for survival/extinction.
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The key to our investigation is an improved (and in a sense sharp) understanding of the survival time of the contact process on star graphs. Using these results, we show that for the contact process on Galton-Watson trees, when the offspring distribution (i) is subexponential the critical value for local survival $lambda_2=0$ and (ii) when it is geometric($p$) we have $lambda_2 le C_p$, where the $C_p$ are much smaller than previous estimates. We also study the critical value $lambda_c(n)$ for prolonged persistence on graphs with $n$ vertices generated by the configuration model. In the case of power law and stretched exponential distributions where it is known $lambda_c(n) to 0$ we give estimates on the rate of convergence. Physicists tell us that $lambda_c(n) sim 1/Lambda(n)$ where $Lambda(n)$ is the maximum eigenvalue of the adjacency matrix. Our results show that this is not correct.
At each site of a supercritical Galton-Watson tree place a parking spot which can accommodate one car. Initially, an independent and identically distributed number of cars arrive at each vertex. Cars proceed towards the root in discrete time and park in the first available spot they come to. Let $X$ be the total number of cars that arrive to the root. Goldschmidt and Przykucki proved that $X$ undergoes a phase transition from being finite to infinite almost surely as the mean number of cars arriving to each vertex increases. We show that $EX$ is finite at the critical threshold, describe its growth rate above criticality, and prove that it increases as the initial car arrival distribution becomes less concentrated. For the canonical case that either 0 or 2 cars arrive at each vertex of a $d$-ary tree, we give improved bounds on the critical threshold and show that $P(X = 0)$ is discontinuous.
We study the totally asymmetric simple exclusion process (TASEP) on trees where particles are generated at the root. Particles can only jump away from the root, and they jump from $x$ to $y$ at rate $r_{x,y}$ provided $y$ is empty. Starting from the all empty initial condition, we show that the distribution of the configuration at time $t$ converges to an equilibrium. We study the current and give conditions on the transition rates such that the current is of linear order or such that there is zero current, i.e. the particles block each other. A key step, which is of independent interest, is to bound the first generation at which the particle trajectories of the first $n$ particles decouple.
A recursive function on a tree is a function in which each leaf has a given value, and each internal node has a value equal to a function of the number of children, the values of the children, and possibly an explicitly specified random element $U$. The value of the root is the key quantity of interest in general. In this first study, all node values and function values are in a finite set $S$. In this note, we describe the limit behavior when the leaf values are drawn independently from a fixed distribution on $S$, and the tree $T_n$ is a random Galton--Watson tree of size $n$.
Distinguishing between continuous and first-order phase transitions is a major challenge in random discrete systems. We study the topic for events with recursive structure on Galton-Watson trees. For example, let $mathcal{T}_1$ be the event that a Galton-Watson tree is infinite, and let $mathcal{T}_2$ be the event that it contains an infinite binary tree starting from its root. These events satisfy similar recursive properties: $mathcal{T}_1$ holds if and only if $mathcal{T}_1$ holds for at least one of the trees initiated by children of the root, and $mathcal{T}_2$ holds if and only if $mathcal{T}_2$ holds for at least two of these trees. The probability of $mathcal{T}_1$ has a continuous phase transition, increasing from 0 when the mean of the child distribution increases above 1. On the other hand, the probability of $mathcal{T}_2$ has a first-order phase transition, jumping discontinuously to a nonzero value at criticality. Given the recursive property satisfied by the event, we describe the critical child distributions where a continuous phase transition takes place. In many cases, we also characterize the event undergoing the phase transition.
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