Topic
Homogeneous tree
About: Homogeneous tree is a research topic. Over the lifetime, 135 publications have been published within this topic receiving 6272 citations.
Papers published on a yearly basis
Papers
TL;DR: In this article, it was shown that the contact process on an infinite homogeneous tree exhibits at least two phase transitions as the infection parameter is varied, for small values of λ, where the infection eventually dies out and for larger λ the infection lives forever with positive probability but eventually leaves any finite set.
Abstract: The contact process on an infinite homogeneous tree is shown to exhibit at least two phase transitions as the infection parameter $\lambda$ is varied. For small values of $\lambda$ a single infection eventually dies out. For larger $\lambda$ the infection lives forever with positive probability but eventually leaves any finite set. (The survival probability is a continuous function of $\lambda$, and the proof of this is much easier than it is for the contact process on $d$-dimensional integer lattices.) For still larger $\lambda$ the infection converges in distribution to a nontrivial invariant measure. For any $n$-ary tree, with $n$ large, the first of these transitions occurs when $\lambda \approx 1/n$ and the second occurs when $1/2\sqrt{n} < \lambda < e/\sqrt{n}$. Nonhomogeneous trees whose vertices have degrees varying between 1 and $n$ behave essentially as homogeneous $n$-ary trees, provided that vertices of degree $n$ are not too rare. In particular, letting $n$ go to $\infty$, Galton-Watson trees whose vertices have degree $n$ with probability that does not decrease exponentially with $n$ may have both phase transitions occur together at $\lambda = 0$. The nature of the second phase transition is not yet clear and several problems are mentioned in this regard.
199 citations
TL;DR: In this paper, the authors studied Markov chains in an infinite homogeneous tree of order a+1 and proved a series of results using essentially the spherical functions of this symmetric space.
Abstract: Let T be an infinite homogeneous tree of order a+1. We study Markov chains {Xn} in T whose transition functions p(x, y)=A[d(x,y)] depend only on the shortest distance between x and y in the graph. The graph T can be represented as a symmetric space of a p-adic matrix group; we prove a series of results using essentially the spherical functions of this symmetric space. Theorem 1.d(Xn,x)∼β n a.s., where β>0 if A(0) ≠ 1, X0=x. Assuming {Xn} is strongly aperiodic, Theorem 2. p2(x, y)∼CRn/n3/2 for fixed x, y where R=∑φ(d) A(d)<1, and if E[d(X1, X0)2]<∞, Theorem 3. R(1−u, x, y) = ∑(1−u)npn(x, y)=Ca−d[exp(−du/β)+od(1)] as d=d(x,y)→∞ uniformly for 0≦u≦2. Using Theorem 3, we calculate the Martin boundary Dirichlet kernel of p(x, y) on T, which turns out to be independent of {itA(d)}. We also consider a “stepping-stone” model of a randomly-mating-and-migrating population on the nodes of T. If initially all individuals are distinct, then in generation n approximately half of the individuals of a given type are within βn of a typical one and essentially all are within 2βn.
115 citations
TL;DR: In this article, it was shown that the contact process on a homogeneous tree exhibits two phase transitions when $d \geq 3, a behavior which distinguishes it from the contact phase on a binary tree.
Abstract: Let $\mathbb{T}_d$ be a homogeneous tree in which every vertex has $d$ neighbors A new proof is given that the contact process on $\mathbb{T}_d$ exhibits two phase transitions when $d \geq 3$, a behavior which distinguishes it from the contact process on $\mathbb{Z}^n$ This is the first proof which does not involve calculation of bounds on critical values, and it is much shorter than the previous proof for the binary tree, $\mathbb{T}_3$ The method is extended to prove the existence of an intermediate phase for a more general class of trees with exponential growth and certain symmetry properties, for which no such result was previously known
100 citations
Posted Content•
TL;DR: In this paper, the contact process on an infinite homogeneous tree is shown to exhibit at least two phase transitions as the infection parameter lambda is varied, for small values of lambda a single infection eventually dies out and for larger lambda the infection lives forever with positive probability but eventually leaves any finite set.
Abstract: The contact process on an infinite homogeneous tree is shown to exhibit at least two phase transitions as the infection parameter lambda is varied. For small values of lambda a single infection eventually dies out. For larger lambda the infection lives forever with positive probability but eventually leaves any finite set. (The survival probability is a continuous function of lambda, and the proof of this is much easier than it is for the contact process on d-dimensional integer lattices.) For still larger lambda the infection converges in distribution to a nontrivial invariant measure. For an n-ary tree, with n large, the first of these transitions occurs when lambda~1/n and the second occurs when 1/2 sqrt{n}
98 citations
18 Dec 1997
TL;DR: In this article, the analysis of Hill's operator D 2 + q(x) for qeven and periodic is extended from the real line to homogeneous trees T. The spectrum is exactly described when the degree of the tree is greater than two, in which case there are both spectral bands and eigenvalues.
Abstract: The analysis of Hill’s operator D 2 + q(x)for qeven and periodic is extended from the real line to homogeneous trees T. Generalizing the classical problem, a detailed analysis of Hill’s equation and its related operatortheoryon L 2 (T)isprovided. Themultipliersforthisnewversion of Hill’s equation are identied and analyzed. An explicit description of theresolventis given. The spectrumis exactly describedwhen thedegree of the tree is greater than two, in which case there are both spectral bands and eigenvalues. Spectral projections are computed by means of an eigenfunction expansion. Long time asymptotic expansions for the associated semigroup kernel are also described. A summation formula expresses the resolvent for a regular graph as a function of the resolvent of its covering homogeneous tree and the covering map. In the case of a nite regular graph, a trace formula relates the spectrum of the Hill’s operator to the lengths of closed paths in the graph.
62 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 10 |
| 2020 | 8 |
| 2019 | 4 |
| 2018 | 4 |
| 2017 | 3 |
| 2016 | 2 |