Sharpness of the phase transition for parking on random trees

具有（随机）给定度序列的树上停车问题与 Frozen 配置模型

Parking on trees with a (random) given degree sequence and the Frozen configuration model

本文通过将给定度序列树上的停车过程与有向配置模型自然耦合，重新推导出临界 Bienaymé–Galton–Watson 树上停车相变的位置。

Parking on a Random Tree

Consider an infinite tree with random degrees, i.i.d. over the sites, with a prescribed probability distribution with generating function G(s). We consider the following variation of Renyi's parking problem, alternatively called blocking RSA: at every vertex of the tree a particle (or car) arrives with rate one. The particle sticks to the vertex whenever the vertex and all of its nearest neighbors are not occupied yet. We provide an explicit expression for the so-called parking constant in terms of the generating function.

超临界几何Bienaymé--Galton--Watson树上的停车问题

Parking on supercritical geometric Bienaymé--Galton--Watson trees

本文针对超临界几何Bienaymé--Galton--Watson树上的停车过程，基于到达分布的生成函数，给出了判定过程处于亚临界、临界或超临界状态的准则。

Sharpness of second moment criteria for branching and tree-indexed processes

A class of branching processes in varying environments is exhibited which become extinct almost surely even though the means M_n grow fast enough so that sum M_n^{-1} is finite. In fact, such a process is constructed for every offspring distribution of infinite variance, and this establishes the converse of a previously known fact: that if a distribution has finite variance then sum M_n^{-1}=infty is equivalent to almost sure extinction. This has as an immediate consequence the converse to a theorem on equipolarity of Galton-Watson trees.

The phase transition for parking on Galton--Watson trees

We establish a phase transition for the parking process on critical Galton--Watson trees. In this model, a random number of cars with mean $m$ and variance $σ^{2}$ arrive independently on the vertices of a critical Galton--Watson tree with finite variance $Σ^{2}$ conditioned to be large. The cars go down the tree towards the root and try to park on empty vertices as soon as possible. We show a phase transition depending on $$ Θ:= (1-m)^2- Σ^2 (σ^2+m^2-m).$$ Specifically, when $m \leq 1$, if $ Θ>0,$ then all but (possibly) a few cars will manage to park, whereas if $Θ<0$, then a positive fracti…

Parking on the infinite binary tree

Let $(A_u : u \in \mathbb{B})$ be i.i.d.~non-negative integers that we interpret as car arrivals on the vertices of the full binary tree $ \mathbb{B}$. Each car tries to park on its arrival node, but if it is already occupied, it drives towards the root and parks on the first available spot. It is known that the parking process on $ \mathbb{B}$ exhibits a phase transition in the sense that either a finite number of cars do not manage to park in expectation (subcritical regime) or all vertices of the tree contain a car and infinitely many cars do not manage to park (supercritical regime). We ch…

Parking on Cayley trees & Frozen Erdös-Rényi

Consider a uniform rooted Cayley tree $T_{n}$ with $n$ vertices and let $m$ cars arrive sequentially, independently, and uniformly on its vertices. Each car tries to park on its arrival node, and if the spot is already occupied, it drives towards the root of the tree and parks as soon as possible. Lackner & Panholzer (arXiv:1504.04972) established a phase transition for this process when $ m \approx \frac{n}{2}$. In this work, we couple this model with a variant of the classical Erdös-Rényi random graph process. This enables us to describe the phase transition for the size of the components of…

A binomial splitting process in connection with corner parking problems

A special type of binomial splitting process is studied. Such a process can be used to model a high-dimensional corner parking problem, as well as the depth of random PATRICIA tries (a special class of digital tree data structures). The later also has natural interpretations in terms of distinct values in iid geometric random variables and the occupancy problem in urn models. The corresponding distribution is marked by logarithmic mean and bounded variance, which is oscillating, if the binomial parameter $p$ is not equal to 1/2, and asymptotic to 1 in the unbiased case. Also, the limiting dist…

Parking on a random tree

Consider a uniform random rooted tree on vertices labelled by $[n] = \{1,2,\ldots,n\}$, with edges directed towards the root. We imagine that each node of the tree has space for a single car to park. A number $m \le n$ of cars arrive one by one, each at a node chosen independently and uniformly at random. If a car arrives at a space which is already occupied, it follows the unique path oriented towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Consider $m =[αn]$ and let $A_{n,α}$ denote the event that all $[αn]$ c…

催化方程与完全停放树的普适性

Universality for catalytic equations and fully parked trees

该研究用概率方法证明完全停放临界树收敛于布朗生长-碎裂树，同时证实含催化变量的正非线性多项式方程奇点处普适指数为5/2的猜想，推进了相关概率与组合领域研究。

Enumeration of fully parked trees

We enumerate a class of fully parked trees. In a probabilistic context, this means computing the partition function $F(x,y)$ of the parking process where an i.i.d. number of cars arrives at each vertex of a Galton-Watson tree with a geometric offspring distribution, conditioned to leave no vertex unoccupied at the end. The variables $x$ and $y$ count the number of vertices in the tree and the number of cars exiting from the root, respectively. For any car arrival distribution $\mathbf b$, we obtain an explicit parametric expression of $F(x,y)$ in terms of the probability generating function $B…