We consider noisy binary channels on regular trees and introduce periodic enhancements consisting of locally self-correcting the signal in blocks without break of the symmetry of the model. We focus on the realistic class of within-descent self-correction realized by identifying all descendants $k$ generations down a vertex with their majority. We show that this also allows reconstruction strictly beyond the critical distortion. We further identify the limit at which the critical distortions of within-descent $k$ self-corrected transmission converge, which turns out to be the critical point for ferromagnetic Ising model on that tree. We finally discuss how similar phenomena take place with the biologically more plausible mechanism of eliminating signals which are locally not coherent with the majority.
The Arboreal gas model on a finite graph $G$ is the Bernoulli bond percolation on $G$ conditioned on the event that the sampled subgraph is a forest. In this short note we study the arboreal gas on a regular tree wired at the leaves and obtain a comprehensive description of the weak limit of this model.
We extend some results of Itai Benjamini and Yuri Lima (see \href{http://arxiv.org/pdf/1305.2610.pdf}{\cite{Benjamini}}). In this paper they consider a binary tree $\mathbb T_n$ of height $n$, each leaf is either infected by one of $k$ diseases or not infected at all. In other words, $x$ at generation $n$ is infected by the $i$-th infection with probability $p_i$ and sane with $p_{k+1}$. Moreover the infections are independently distributed for each leaf. Infections spread along the tree based on specific rules. In their paper they study the limit distribution of the root of $\mathbb T_n$ as $…
We estimate the critical temperature of a family of quantum spin systems on regular trees of large degree. The systems include the spin-$\frac12$ XXZ model and the spin-1 nematic model. Our formula is conjectured to be valid for large-dimensional cubic lattices. Our method of proof uses a probabilistic representation in terms of random loops.
We present a proof of the power law decay of magnetic moment for the $q$-state antiferromagnetic Potts model on the regular tree at the critical temperature, and also justify that the exact exponent is $\frac{1}{2}$. Our proof relies on the assumption of the uniqueness at the critical temperature, which has been established for $q=3,4$, and for $q \ge 5$ with large degree. An iterative contraction inequality is developed for independent interests.
In Bernoulli bond percolation on the Cartesian product graph of a $d$-regular tree and a line, we give an upper bound for the critical probability $p_c$.
In the context of percolation in a regular tree, we study the size of the largest cluster and the length of the longest run starting within the first d generations. As d tends to infinity, we prove almost sure and weak convergence results.
We consider Bernoulli bond percolation on the product graph of a regular tree and a line. We show that the triangle condition does not hold at the uniqueness threshold.
We consider a recurrent random walk in random environment on a regular tree. Under suitable general assumptions upon the distribution of the environment, we show that the walk exhibits an unusual slow movement: the order of magnitude of the walk in the first $n$ steps is $(\log n)^3$.
Consider a vertex-reinforced jump process defined on a regular tree, where each vertex has exactly $b$ children, with $b \ge 3$. We prove the strong law of large numbers and the central limit theorem for the distance of the process from the root. Notice that it is still unknown if vertex-reinforced jump process is transient on the binary tree.
A Markov tree is a random vector indexed by the nodes of a tree whose distribution is determined by the distributions of pairs of neighbouring variables and a list of conditional independence relations. Upon an assumption on the tails of the Markov kernels associated to these pairs, the conditional distribution of the self-normalized random vector when the variable at the root of the tree tends to infinity converges weakly to a random vector of coupled random walks called tail tree. If, in addition, the conditioning variable has a regularly varying tail, the Markov tree satisfies a form of one…
This paper studies the boundary behaviour of $λ$-polyharmonic functions for the simple random walk operator on a regular tree, where $λ$ is complex and $|λ|> ρ$, the $\ell^2$-spectral radius of the random walk. In particular, subject to normalisation by spherical, resp. polyspherical functions, Dirichlet and Riquier problems at infinity are solved and a non-tangential Fatou theorem is proved.
For the Gaussian free field on a $(d + 1)$-regular tree with $d \geq 2$, we study the percolative properties of its level sets in the critical and the near-critical regime. In particular, we show the continuity of the percolation probability, derive an exact symptotic tail estimate for the cardinality of the connected component of the critical level set, and describe the asymptotic behaviour of the percolation probability in the near-critical regime.
In invasion percolation, the edges of successively maximal weight (the outlets) divide the invasion cluster into a chain of ponds separated by outlets. On the regular tree, the ponds are shown to grow exponentially, with law of large numbers, central limit theorem and large deviation results. The tail asymptotics for a fixed pond are also studied and are shown to be related to the asymptotics of a critical percolation cluster, with a logarithmic correction.
It is known that the Kesten-Stigum reconstruction bound is tight for roughly symmetric binary channels. In this paper, we will adopt a refined analysis of moment recursion on a weighted version of the magnetization, which is engaged in Sly [2011] to handle the symmetric Potts model, and establish the critical condition of the asymmetric Ising model to make Kesten-Stigum bound the reconstruction threshold on regular $d$-ary trees.
We consider Bernoulli bond percolation on the product graph of a regular tree and a line. Schonmann showed that there are a.s. infinitely many infinite clusters at $p=p_u$ by using a certain function $α(p)$. The function $α(p)$ is defined by a exponential decay rate of probability that two vertices of the same layer are connected. We show the critical probability $p_c$ can be written by using $α(p)$. In other words, we construct another definition of the critical probability.
We give a motivation and gentle introduction into the regularity structure and model introduced by Otto, Sauer, Smith and Weber, which fall into the framework of Hairer, but have a greedier index set than the one given by trees. We do this here for a simple quasi-linear parabolic equation and assume that the driving noise is so regular that no renormalization is needed. We introduce the abstract model space $\mathsf{T}$ and its grading, the pre-model $\mathbfΠ$, the centered model $Π_x$, the structure group $\mathsf{G}$, and the re-centering transformations $Γ_{xy}$. Using integration and reco…
We study the weak limit of the arboreal gas along any exhaustion of a regular tree with wired boundary conditions. We prove that this limit exists, does not depend on the choice of exhaustion, and undergoes a phase transition. Below and at criticality, we prove the model is equivalent to bond percolation. Above criticality, we characterise the model as the superposition of critical bond percolation and a random collection of infinite one-ended paths. This provides a simple example of an arboreal gas model that continues to exhibit critical-like behaviour throughout its supercritical phase.
We study Random Walks in an i.i.d. Random Environment (RWRE) defined on $b$-regular trees. We prove a functional central limit theorem (FCLT) for transient processes, under a moment condition on the environment. We emphasize that we make no uniform ellipticity assumptions. Our approach relies on regenerative levels, i.e. levels that are visited exactly once. On the way, we prove that the distance between consecutive regenerative levels have a geometrically decaying tail. In the second part of this paper, we apply our results to Linearly Edge-Reinforced Random Walk (LERRW) to prove FCLT when th…
We consider exclusion processes on a rooted $d$-regular tree. We start from a Bernoulli product measure conditioned on having a particle at the root, which we call the tagged particle. For $d\geq 3$, we show that the tagged particle has positive linear speed and satisfies a central limit theorem. We give an explicit formula for the speed. As a key step in the proof, we first show that the exclusion process "seen from the tagged particle" has an ergodic invariant measure.
We prove existence of the scaling limit of the invasion percolation cluster (IPC) on a regular tree. The limit is a random real tree with a single end. The contour and height functions of the limit are described as certain diffusive stochastic processes. This convergence allows us to recover and make precise certain asymptotic results for the IPC. In particular, we relate the limit of the rescaled level sets of the IPC to the local time of the scaled height function.
We consider a branching random walk with binary state space and index set $T^k$, the infinite rooted tree in which each node has k children (also known as the model of "broadcasting on a tree"). The root of the tree takes a random value 0 or 1, and then each node passes a value independently to each of its children according to a 2x2 transition matrix P. We say that "reconstruction is possible" if the values at the d'th level of the tree contain non-vanishing information about the value at the root as $d\to\infty$. Adapting a method of Brightwell and Winkler, we obtain new conditions under whi…
We study the invariant measures of infinite systems of stochastic differential equations (SDEs) indexed by the vertices of a regular tree. These invariant measures correspond to Gibbs measures associated with certain continuous specifications, and we focus specifically on those measures which are homogeneous Markov random fields. We characterize the joint law at any two adjacent vertices in terms of a new two-dimensional SDE system, called the "local equation", which exhibits an unusual dependence on a conditional law. Exploiting an alternative characterization in terms of an eigenfunction-typ…
Consider a statistical physical model on the $d$-regular infinite tree $T_{d}$ described by a set of interactions $Φ$. Let $\{G_{n}\}$ be a sequence of finite graphs with vertex sets $V_n$ that locally converge to $T_{d}$. From $Φ$ one can construct a sequence of corresponding models on the graphs $G_n$. Let $\{μ_n\}$ be the resulting Gibbs measures. Here we assume that $\{μ_{n}\}$ converges to some limiting Gibbs measure $μ$ on $T_{d}$ in the local weak$^*$ sense, and study the consequences of this convergence for the specific entropies $|V_n|^{-1}H(μ_n)$. We show that the limit supremum of $…
In this paper we are concerned with bias voter models on trees and lattices, where the vertex in state 0 reconsiders its opinion at a larger rate than that of the vertex in state 1. For the process on tree with product measure as initial distribution, we obtain a mean field limit at each moment of the probability that a given vertex is in state 1 as the degree of the tree grows to infinity. Furthermore, for our model on trees and lattices, we show that the process converges weakly to the configuration where all the vertices are in state 1 when the rate at which a vertex in state 0 reconsiders …
We study various classes of random processes defined on the regular tree $T_d$ that are invariant under the automorphism group of $T_d$. Most important ones are factor of i.i.d. processes (randomized local algorithms), branching Markov chains and a new class that we call typical processes. Using Glauber dynamics on processes we give a sufficient condition for a branching Markov chain to be factor of i.i.d. Typical processes are defined in a way that they create a correspondence principle between random $d$-reguar graphs and ergodic theory on $T_d$. Using this correspondence principle together …
We study the problem of identifying the source of a diffusion spreading over a regular tree. When the degree of each node is at least three, we show that it is possible to construct confidence sets for the diffusion source with size independent of the number of infected nodes. Our estimators are motivated by analogous results in the literature concerning identification of the root node in preferential attachment and uniform attachment trees. At the core of our proofs is a probabilistic analysis of Pólya urns corresponding to the number of uninfected neighbors in specific subtrees of the infect…
