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Register a new userThe sum of powers of subtree sizes for conditioned Galton-Watson trees
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We study the additive functional $X_n(alpha)$ on conditioned Galton-Watson trees given, for arbitrary complex $alpha$, by summing the $alpha$th power of all subtree sizes. Allowing complex $alpha$ is advantageous, even for the study of real $alpha$, since it allows us to use powerful results from the theory of analytic functions in the proofs. For $Realpha < 0$, we prove that $X_n(alpha)$, suitably normalized, has a complex normal limiting distribution; moreover, as processes in $alpha$, the weak convergence holds in the space of analytic functions in the left half-plane. We establish, and prove similar process-convergence extensions of, limiting distribution results for $alpha$ in various regions of the complex plane. We focus mainly on the case where $Realpha > 0$, for which $X_n(alpha)$, suitably normalized, has a limiting distribution that is not normal but does not depend on the offspring distribution $xi$ of the conditioned Galton-Watson tree, assuming only that $E[xi] = 1$ and $0 < mathrm{Var} [xi] < infty$. Under a weak extra moment assumption on $xi$, we prove that the convergence extends to moments, ordinary and absolute and mixed, of all orders. At least when $Realpha > frac12$, the limit random variable $Y(alpha)$ can be expressed as a function of a normalized Brownian excursion.
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We consider multi-type Galton Watson trees, and find the distribution of these trees when conditioning on very general types of recursive events. It turns out that the conditioned tree is again a multi-type Galton Watson tree, possibly with more types and with offspring distributions, depending on the type of the father node and on the height of the father node. These distributions are given explicitly. We give some interesting examples for the kind of conditioning we can handle, showing that our methods have a wide range of applications.
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.
When normal and mis`{e}re games are played on bi-type binary Galton-Watson trees (with vertices coloured blue or red and each having either no child or precisely $2$ children), with one player allowed to move along monochromatic edges and the other along non-monochromatic edges, the draw probabilities equal $0$ unless every vertex gives birth to one blue and one red child. On bi-type Poisson trees where each vertex gives birth to Poisson$(lambda)$ offspring in total, the draw probabilities approach $1$ as $lambda rightarrow infty$. We study such emph{nove
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.
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