Cover time for the frog model on trees

Infection spread for the frog model on trees

The frog model is an infection process in which dormant particles begin moving and infecting others once they become infected. We show that on the rooted $d$-ary tree with particle density $Ω(d^2)$, the set of visited sites contains a linearly expanding ball and the number of visits to the root grows linearly with high probability.

The frog model on non-amenable trees

We examine an interacting particle system on trees commonly referred to as the frog model. For its initial state, it begins with a single active particle at the root and i.i.d. $\mathrm{Poiss}(λ)$ many inactive particles at each non-root vertex. Active particles perform discrete time simple random walk and in the process activate any inactive particles they encounter. We show that for $\textit{every}$ non-amenable tree with bounded degree there exists a phase transition from transience to recurrence (with a non-trivial intermediate phase sometimes sandwiched in between) as $λ$ varies.

Critical parameter of the frog model on homogeneous trees with geometric lifetime

We consider the frog model with geometric lifetime (parameter $1-p$) on homogeneous trees of dimension $d$. In 2002, \cite{alves2002-2} proved that there exists a critical lifetime parameter $p_c\in(0,1)$ above which infinitely many frogs are activated with positive probability, and they gave lower and upper bounds for $p_c$. Since then, the literature on this model focussed on refinements of the upper bound. In the present paper we improve the bounds for $p_c$ \emph{on both sides}. We also provide a discussion comparing the bounds of the literature and their proofs. Our proofs are based on co…

A new upper bound for the critical probability of the frog model on homogeneous trees

We consider the interacting particle system on the homogeneous tree of degree $(d + 1)$, known as frog model. In this model, active particles perform independent random walks, awakening all sleeping particles they encounter, and dying after a random number of jumps, with geometric distribution. We prove an upper bound for the critical parameter of survival of the model, which improves the previously known results. This upper bound was conjectured in a paper by Lebensztayn et al. ($ J. Stat. Phys.$, 119(1-2), 331-345, 2005). We also give a closed formula for the upper bound.

From transience to recurrence with Poisson tree frogs

Consider the following interacting particle system on the $d$-ary tree, known as the frog model: Initially, one particle is awake at the root and i.i.d. Poisson many particles are sleeping at every other vertex. Particles that are awake perform simple random walks, awakening any sleeping particles they encounter. We prove that there is a phase transition between transience and recurrence as the initial density of particles increases, and we give the order of the transition up to a logarithmic factor.

Galton-Watson树上的青蛙模型

The frog model on Galton-Watson trees

本文研究Galton-Watson树上的青蛙模型，证明在后代分布满足相应条件时存在区分常返与瞬态的有限正临界值λ_c，且该值依赖完整后代分布。

Phase transition for the frog model on biregular trees

We study the frog model with death on the biregular tree $\mathbb{T}_{d_1,d_2}$. Initially, there is a random number of awake and sleeping particles located on the vertices of the tree. Each awake particle moves as a discrete-time independent simple random walk on $\mathbb{T}_{d_1,d_2}$ and has a probability of death $(1-p)$ before each step. When an awake particle visits a vertex which has not been visited previously, the sleeping particles placed there are awakened. We prove that this model undergoes a phase transition: for values of $p$ below a critical probability $p_c$, the system dies ou…

再探带死亡的青蛙模型

The frog model with death revisited

研究$d$叉树上带死亡与漂移的青蛙模型，证明其常返概率可取正值，从而关于漂移参数不具有单调性。

Recurrence and transience for the frog model on trees

The frog model is a growing system of random walks where a particle is added whenever a new site is visited. A longstanding open question is how often the root is visited on the infinite $d$-ary tree. We prove the model undergoes a phase transition, finding it recurrent for $d=2$ and transient for $d\geq 5$. Simulations suggest strong recurrence for $d=2$, weak recurrence for $d=3$, and transience for $d\geq 4$. Additionally, we prove a 0-1 law for all $d$-ary trees, and we exhibit a graph on which a 0-1 law does not hold. To prove recurrence when $d=2$, we construct a recursive distributional…

Critical drift estimates for the frog model on trees

Place an active particle at the root of a $d$-ary tree and a single dormant particle at each non-root site. In discrete time, active particles move towards the root with probability $p$ and, otherwise, away from the root to a uniformly sampled child vertex. When an active particle moves to a site containing a dormant particle, the dormant particle becomes active. The critical drift $p_d$ is the infimum over all $p$ for which infinitely many particles visit the root almost surely. Guo, Tang, and Wei proved that $\sup_{d\geq 3} p_d \leq 1/3$. We improve this bound to $5/17$ with a shorter argume…

树上青蛙模型的改进临界漂移估计

Improved critical drift estimates for the frog model on trees

本文研究无限 $d$ 叉树上青蛙模型的临界漂移，改进了 $\sup_{d\ge m}p_d$ 的界，并证明自相似变体对应临界值的单调性。

青蛙模型的相变

Phase transition for the frog model

本文研究了图上的简单随机游动系统，称为青蛙模型。这个模型可以描述为：在某个图G上存在活动粒子和睡眠粒子。每个活动粒子执行离散时间的简单随机游动，并且在每个时刻它可能以概率1-p消失。当活动粒子撞击睡眠粒子时，后者变得活跃。给出了Z^d树和规则树的相变结果和临界参数的渐近值。

A zero-one law for recurrence and transience of frog processes

We provide sufficient conditions for the validity of a dichotomy, i.e. zero-one law, between recurrence and transience of general frog models. In particular, the results cover frog models with i.i.d. numbers of frogs per site where the frog dynamics are given by quasi-transitive Markov chains or by random walks in a common random environment including super-critical percolation clusters on $\mathbb{Z}^d$. We also give a sufficient and almost sharp condition for recurrence of uniformly elliptic frog processes on $\mathbb{Z}^d$. Its proof uses the general zero-one law.

The coverage ratio of the frog model on complete graphs

The frog model is a system of interacting random walks. Initially, there is one particle at each vertex of a connected graph $\mathcal{G}$. All particles are inactive at time zero, except for the one which is placed at the root of $\mathcal{G}$, which is active. At each instant of time, each active particle may die with probability $1-p$. Once an active particle survives, it jumps on one of its nearest vertices, chosen with uniform probability, performing a discrete time simple symmetric random walk (SRW) on $\mathcal{G}$. Up to the time it dies, it activates all inactive particles it hits alo…

Frogs on trees?

We study a system of simple random walks on $\mathcal{T}_{d,n} = \mathcal{V}_{d,n}, \mathcal{E}_{d,n})$, the $d$-ary tree of depth $n$, known as the frog model. Initially there are Pois($λ$) particles at each site, independently, with one additional particle planted at some vertex $\mathbf{o}$. Initially all particles are inactive, except for the ones which are placed at $\mathbf{o}$. Active particles perform (independent) $ t \in \mathbb{N} \cup \{\infty \} $ steps of simple random walk on the tree. When an active particle hits an inactive particle, the latter becomes active. The model is oft…

Frog Model覆盖完全图的临界条件

Critical Conditions for the Coverage of Complete Graphs with the Frog Model

本文研究frog model在完全图上的覆盖问题，确定了当粒子存活概率$p_n$趋近于1时图被高概率覆盖的条件，并给出了使所有顶点被访问的临界增长率$p_n=1-α/\log n$。

d≥2时Z^d上青蛙模型首达时间的上尾大偏差

Upper tail large deviation for the first passage time in the frog model on $Z^d$ with $d\geq 2$

该论文研究d≥2时Z^d上青蛙模型首达时间的上尾大偏差，推导了大偏差渐近公式，揭示终点附近的减速机制，通过修正探索过程、重正化与覆盖论证完成机制分析。

一维青蛙模型的上尾大偏差

Upper tail large deviation for the one-dimensional frog model

本文研究一维青蛙模型首达时间的上尾大偏差，提出新的覆盖过程方法，证明其大偏差速率为 $n^{1/2}$，并给出由布朗运动性质刻画的严格凹显式速率函数。

重尾青蛙模型

The Heavy-tailed Frog Model

该研究针对重尾转移核下的青蛙模型，明确了不同尾指数区间的激活时间尺度，并揭示了有限寿命模型的覆盖寿命渐近特性，丰富了非经典随机游走驱动的传播过程理论。

On the cover time of λ-biased walk on supercritical Galton-Watson trees

In this paper, we study the time required for a λ-biased (λ>1) walk to visit all the vertices of a supercritical Galton-Watson tree up to generation n. Inspired by the extremal landscape approach in [Cortines, Louidor, Saglietti 2018] for the simple random walk on binary trees, we establish the scaling limit of the cover time in the biased setting.

二叉树覆盖时间的一个精确估计

A sharp estimate for cover times on binary trees

我们计算的覆盖时间的二叉树深度$n$（连续时间）随机游动的二阶校正，并显示概率接近1的$n$增加，$\sqrt{τ_{\mathrm{cov}}}=\sqrt{|E|}[\sqrt{2\log 2}\cdot n - {\log n}/{\sqrt{2\log 2}} + O((\log\logn)^8]$，从而表明，二阶校正不同于相应的树上的高斯自由场的最大值。

高斯自由域上覆盖时间的渐近性：有界度图和一般树

Asymptotics of cover times via Gaussian free fields: Bounded-degree graphs and general trees

本文证明了在有界度图和一般树上，简单随机游动的覆盖时间渐近等于边数与图上高斯自由场期望上确界的平方的乘积，并假定最大首中时明显小于覆盖时间.以前，这只证明了正规树和2D格。此外，对于一般的树，我们得到指数集中的覆盖时间，这意味着覆盖时间的标准差是有界的覆盖时间和最大首中时的几何平均。

Limit law for the cover time of a random walk on a binary tree

Let $T_n$ denote the binary tree of depth $n$ augmented by an extra edge connected to its root. Let $C_n$ denote the cover time of $T_n$ by simple random walk. We prove that $\sqrt{ \mathcal{C}_{n} 2^{-(n+1) } } - m_n$ converges in distribution as $n\to \infty$, where $m_n$ is an explicit constant, and identify the limit.

A Scaling Limit for the Cover Time of the Binary Tree

We consider a continuous time random walk on the rooted binary tree of depth $n$ with all transition rates equal to one and study its cover time, namely the time until all vertices of the tree have been visited. We prove that, normalized by $2^{n+1} n$ and then centered by $(\log 2) n - \log n$, the cover time admits a weak limit as the depth of the tree tends to infinity. The limiting distribution is identified as that of a Gumbel random variable with rate one, shifted randomly by the logarithm of the sum of the limits of the derivative martingales associated with two negatively correlated di…

Barrier estimates for a critical Galton--Watson process and the cover time of the binary tree

For the critical Galton--Watson process with geometric offspring distributions we provide sharp barrier estimates for barriers which are (small) perturbations of linear barriers. These are useful in analyzing the cover time of finite graphs in the critical regime by random walk, and the Brownian cover times of compact two dimensional manifolds. As an application of the barrier estimates, we prove that if $C_L$ denotes the cover time of the binary tree of depth $L$ by simple walk, then $\sqrt{C_L/2^{L+1}} -\sqrt{2\log 2} L+\log L/\sqrt{2\log 2}$ is tight. The latter improves results of Aldous (…

基于树的布朗交、覆盖时间和厚点

Brownian intersections, cover times and thick points via trees

树上的布朗运动路径的交点与渗流有着密切的联系。最近，来自树上概率的思想是布朗占据测度的多重分形分析的一个重要组成部分。Dembo，J. Rosen和O. Zeitouni因此，我们证明了两个关于二维简单随机游动的定理：第一个定理是Erdans和Taylor（1960）提出的，它涉及在游动的前$n$步中访问最多格点的次数。第二个是Aldous（1989）提出的，它涉及到用$n$格环覆盖$n$的所有点所需的简单随机游动的步数。讲座的目标是如何从树的概率方法可以应用到随机游动和布朗运动在欧几里得空间。

The Green's function on the double cover of the grid and application to the uniform spanning tree trunk

We compute the Green's function on the double cover of ${\mathbb Z}^2$, branched over a vertex or a face. We use this result to compute the local statistics of the "trunk" of the uniform spanning tree on the square lattice, i.e., the limiting probabilities of cylinder events conditional on the path connecting far away points passing through a specified edge. We also show how to compute the local statistics of large-scale triple points of the uniform spanning tree, where the trunk branches. The method reduces the problem to a dimer system with isolated monomers, and we compute the inverse Kaste…

布朗连续随机树上的布朗运动的覆盖时间

On the cover time of Brownian motion on the Brownian continuum random tree

本文证明了布朗连续随机树上的布朗运动的覆盖时间等于局部时间处处严格为正的时刻的下确界，并利用递归自相似性和新的Ray-Knight定理，推导出临界有限方差Galton-Watson树上简单随机游走的覆盖时间在退火律下依分布收敛到该覆盖时间。

Cover time for branching random walks on regular trees

Let $T$ be the regular tree in which every vertex has exactly $d\ge 3$ neighbours. Run a branching random walk on $T$, in which at each time step every particle gives birth to a random number of children with mean $d$ and finite variance, and each of these children moves independently to a uniformly chosen neighbour of its parent. We show that, starting with one particle at some vertex $0$ and conditionally on survival of the process, the time it takes for every vertex within distance $r$ of $0$ to be hit by a particle of the branching random walk is almost surely $r + \frac{2}{\log(3/2)}\log\…

深度为 $n$ 的二叉树上 $λ<1$ 的 $λ$-偏置随机游走覆盖时间的标度极限

Scaling limit for the cover time of the $λ$-biased random walk on a binary tree with $λ<1$

本文研究了深度为 $n$ 的二叉树上 $λ<1$ 的 $λ$-偏置随机游走，推导出其覆盖时间的 $n\rightarrow \infty$ 标度极限，该分布极限由 Cantor 集上的跳跃过程描述，并补充了 $λ\geq 1$ 时的已有结果。

树木覆盖时间的确定性近似

Deterministic approximation for the cover time of trees

我们提出了一个确定性算法，给定一个有n个顶点的树T，一个起始顶点v和一个松弛参数λ> 0，在λ的加性误差内估计覆盖和返回时间，即从v开始的简单随机游走访问T的所有顶点并返回到v所需的期望时间。我们的算法的运行时间是n/λ的多项式，因此，对于n = 1/n^{O（1）}，仍然是n中的多项式。我们还展示了该算法如何可以扩展到估计预期的覆盖（不返回）时间的树木。

图的有向覆盖上Rotor-Router游动的瞬时性和常返性

Transience and recurrence of rotor-router walks on directed covers of graphs

本文的目的是推广Angel和Holroyd关于超限转子-路由器游动的瞬时性和常返性的结果，对于均匀树上转子的随机初始构形。我们在有限图的有向覆盖上解决了同样的问题，这些图也被称为具有多个锥类型的树或周期树。此外，我们提供了一个有向覆盖的例子，使得转子路由器行走可以是递归的或瞬态的，这仅取决于周期树的平面嵌入。