Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userExistence of absolutely continuous spectrum for Galton-Watson random trees
103
0
0.0
(
0
)
Authors
Adam Arras
Ask ChatGPT about the research
No Arabic abstract
We establish a quantitative criterion for an operator defined on a Galton-Watson random tree for having an absolutely continuous spectrum. For the adjacency operator, this criterion requires that the offspring distribution has a relative variance below a threshold. As a byproduct, we prove that the adjacency operator of a supercritical Poisson Galton-Watson tree has a non-trivial absolutely continuous part if the average degree is large enough. We also prove that its Karp and Sipser core has purely absolutely spectrum on an interval if the average degree is large enough. We finally illustrate our criterion on the Anderson model on a d-regular infinite tree with d $ge$ 3 and give a quantitative version of Kleins Theorem on the existence of absolutely continuous spectrum at disorder smaller that C $sqrt$ d for some absolute constant C.
rate research
Read More
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.
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 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.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
