Cutoff on trees is rare

Branching random walks conditioned on rarely survival

In this paper, we show that a Galton-Watson tree conditioned to have a fixed number of particles in generation $n$ converges in distribution as $n\rightarrow\infty$, and with this tool we study the span and gap statistics of a branching random walk on such trees, which is the discrete version of Ramola, Majumdar and Schehr 2015, generalized to arbitrary offspring and displacement distributions with moment constraints.

大小条件 Bienaymé 树中的稀有子树模式：泊松近似与解聚

Rare subtree patterns in size-conditioned Bienaymé trees: Poisson approximation and declumping

研究临界 Bienaymé-Galton-Watson 树中稀有子树模式，结合多种方法克服大小条件依赖性，建立泊松近似，应用于最大叶高、完全$r$元树高度等，给出不同情况下的渐近结果及行为特点。

Asymptotically Optimal Importance Sampling for Jackson Networks with a Tree Topology

Importance sampling (IS) is a variance reduction method for simulating rare events. A recent paper by Dupuis, Wang and Sezer (Ann. App. Probab. 17(4):1306- 1346, 2007) exploits connections between IS and stochastic games and optimal control problems to show how to design and analyze simple and efficient IS algorithms for various overflow events for tandem Jackson networks. The present paper uses the same approach to build asymptotically optimal IS schemes for stable open Jackson networks with a tree topology. Customers arrive at the single root of the tree. The rare overflow event we consider …

Supercritical percolation on large scale-free random trees

We consider Bernoulli bond percolation on a large scale-free tree in the supercritical regime, meaning informally that there exists a giant cluster with high probability. We obtain a weak limit theorem for the sizes of the next largest clusters, extending a recent result for large random recursive trees. The approach relies on the analysis of the asymptotic behavior of branching processes subject to rare neutral mutations, which may be of independent interest.

Multi-time scales in adaptive dynamics: microscopic interpretation of a trait substitution tree model

We consider a fitness-structured population model with competition and migration between nearest neighbors. Under a combination of large population and rare migration limits we are particularly interested in the asymptotic behavior of the total population partition on supporting trait sites. For the population without mutation on a finite trait space we obtain the equilibrium configuration and characterize the right time scale for fixation. For the model with mutation on an infinite trait space a jump process-trait substitution tree model is established on a rarer mutation time scale against t…

Random walks on dynamic configuration models: a trichotomy

We consider a dynamic random graph on $n$ vertices that is obtained by starting from a random graph generated according to the configuration model with a prescribed degree sequence and at each unit of time randomly rewiring a fraction $α_n$ of the edges. We are interested in the mixing time of a random walk without backtracking on this dynamic random graph in the limit as $n\to\infty$, when $α_n$ is chosen such that $\lim_{n\to\infty} α_n (\log n)^2 = β\in [0,\infty]$. In [1] we found that, under mild regularity conditions on the degree sequence, the mixing time is of order $1/\sqrt{α_n}$ when…

Scaling limits of Markov branching trees with applications to Galton-Watson and random unordered trees

We consider a family of random trees satisfying a Markov branching property. Roughly, this property says that the subtrees above some given height are independent with a law that depends only on their total size, the latter being either the number of leaves or vertices. Such families are parameterized by sequences of distributions on partitions of the integers that determine how the size of a tree is distributed in its different subtrees. Under some natural assumption on these distributions, stipulating that "macroscopic" splitting events are rare, we show that Markov branching trees admit the…

Random walks on the random graph

We study random walks on the giant component of the Erdős-Rényi random graph ${\cal G}(n,p)$ where $p=λ/n$ for $λ>1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald, to have order $\log^2 n$. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to $O(\log n)$ and concentrates it (the cutoff phenomenon occurs): the typical mixing is at $(ν{\bf d})^{-1}\log n \pm (\log n)^{1/2+o(1)}$, where $ν$ and ${\bf d}$ are the speed …

Comparing mixing times on sparse random graphs

It is natural to expect that nonbacktracking random walk will mix faster than simple random walks, but so far this has only been proved in regular graphs. To analyze typical irregular graphs, let $G$ be a random graph on $n$ vertices with minimum degree 3 and a degree distribution that has exponential tails. We determine the precise worst-case mixing time for simple random walk on $G$, and show that, with high probability, it exhibits cutoff at time $\mathbf{h}^{-1} \log n$, where $\mathbf{h}$ is the asymptotic entropy for simple random walk on a Galton--Watson tree that approximates $G$ local…