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Register a new userConditioned multi-type Galton-Watson trees
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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.
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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 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.
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.
We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than $eps$, agrees up to generation $K$ with a regular $mu$-ary tree, where $mu$ is the essential minimum of the offspring distribution and the random variable $K$ is strongly concentrated near an explicit deterministic function growing like a multiple of $log(1/eps)$. More precisely, we show that if $muge 2$ then with high probability as $eps downarrow 0$, $K$ takes exactly one or two values. This shows in particular that the conditioned trees converge to the regular $mu$-ary tree, providing an example of entropic repulsion where the limit has vanishing entropy.
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