Related papers: Parking on transitive unimodular graphs
Models of parking in which cars are placed randomly and then move according to a deterministic rule have been studied since the work of Konheim and Weiss in the 1960s. Recently, Damron, Gravner, Junge, Lyu, and Sivakoff introduced a model…
In this paper, we investigate a parking process on a uniform random rooted plane tree with $n$ vertices. Every vertex of the tree has a parking space for a single car. Cars arrive at independent uniformly random vertices of the tree. If the…
In this paper, we investigate the parking process on a uniform random rooted binary tree with $n$ vertices. Viewing each vertex as a single parking space, a random number of cars independently arrive at and attempt to park on each vertex…
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…
Initially a car is placed with probability p at each site of the two-dimensional integer lattice. Each car is equally likely to be East-facing or North-facing, and different sites receive independent assignments. At odd time steps, each…
In parking problems, a given number of cars enter a one-way street sequentially, and try to park according to a specified preferred spot in the street. Various models are possible depending on the chosen rule for collisions, when two cars…
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…
We investigate parking in a one-dimensional lot, where cars enter at a rate $\lambda$ and each attempts to park close to a target at the origin. Parked cars also depart at rate 1. An entering driver cannot see beyond the parked cars for…
Suppose that $m$ drivers each choose a preferred parking space in a linear car park with $n$ spots. In order, each driver goes to their chosen spot and parks there if possible, and otherwise takes the next available spot if it exists. If…
In this paper, the parking problem of a swarm of mobile robots has been studied. The robots are deployed at the nodes of an infinite grid, which has a subset of prefixed nodes marked as parking nodes. Each parking node p_i has a capacity of…
We introduce the notion of a "random basic walk" on an infinite graph, give numerous examples, list potential applications, and provide detailed comparisons between the random basic walk and existing generalizations of simple random walks.…
Consider simple random walk $(X_n)_{n\geq0}$ on a transitive graph with spectral radius $\rho$. Let $u_n=\mathbb{P}[X_n=X_0]$ be the $n$-step return probability and $f_n$ be the first return probability at time $n$. It is a folklore…
Coalescing random walk on a unimodular random rooted graph for which the root has finite expected degree visits each site infinitely often almost surely. A corollary is that an opinion in the voter model on such graphs has infinite expected…
Parking functions correspond with preferences of $n$ cars which enter sequentially to park on a one-way street where (1) each car parks in the first available spot greater than or equal to its preference and (2) all cars successfully park.…
Suppose that $m$ drivers each choose a preferred parking space in a linear car park with $n$ spots. In order, each driver goes to their chosen spot and parks there if possible, and otherwise takes the next available spot if it exists. If…
We study the asymptotic behavior of the number of paths of length $N$ on several classes of infinite graphs with a single special vertex. This vertex can work as an entropic trap for the path, i.e. under certain conditions the dominant part…
Let $G$ be a graph. Assume that to each vertex of a set of vertices $S\subseteq V(G)$ a robot is assigned. At each stage one robot can move to a neighbouring vertex. Then $S$ is a mobile general position set of $G$ if there exists a…
For a labeled, rooted tree with edges oriented towards the root, we consider the vertices as parking spots and the edge orientation as a one-way street. Each driver, starting with her preferred parking spot, searches for and parks in the…
We prove that for the Activated Random Walks model on transitive unimodular graphs, if there is fixation, then every particle eventually fixates, almost surely. We deduce that the critical density is at most 1. Our methods apply for much…
We study the parking process on the random recursive tree. We first prove that although the random recursive tree has a non-degenerate Benjamini--Schramm limit, the phase transition for the parking process appears at density $0$. We then…