Consider an infinite graph with nodes initially labeled by independent Bernoulli random variables of parameter p. We address the density classification problem, that is, we want to design a (probabilistic or deterministic) cellular automaton or a finite-range interacting particle system that evolves on this graph and decides whether p is smaller or larger than 1/2. Precisely, the trajectories should converge to the uniform configuration with only 0's if p<1/2, and only 1's if p>1/2. We present solutions to that problem on the d-dimensional lattice, for any d>1, and on the regular infinite trees. For Z, we propose some candidates that we back up with numerical simulations.
We consider Galton-Watson trees with Geom$(p)$ offspring distribution. We let $T_{\infty}(p)$ denote such a tree conditioned on being infinite. We prove that for any $1/2\leq p_1 <p_2 \leq 1$, there exists a coupling between $T_{\infty}(p_1)$ and $T_{\infty}(p_2)$ such that ${\mathbb P}(T_{\infty}(p_1) \subseteq T_{\infty}(p_2))=1.$
Let $T$ be an infinite rooted tree with weights $w_e$ assigned to its edges. Denote by $m_n(T)$ the minimum weight of a path from the root to a node of the $n$th generation. We consider the possible behaviour of $m_n(T)$ with focus on the two following cases: we say $T$ is explosive if \[ \lim_{n\to \infty}m_n(T) < \infty, \] and say that $T$ exhibits linear growth if \[ \liminf_{n\to \infty} \frac{m_n(T)}{n} > 0. \] We consider a class of infinite randomly weighted trees related to the Poisson-weighted infinite tree, and determine precisely which trees in this class have linear growth almost …
We consider Galton-Watson trees with ${\rm Bin}(d,p)$ offspring distribution. We let $T_{\infty}(p)$ denote such a tree conditioned on being infinite. For $d=2,3$ and any $1/d\leq p_1 <p_2 \leq 1$, we show that there exists a coupling between $T_{\infty}(p_1)$ and $T_{\infty}(p_2)$ such that ${\mathbb P}(T_{\infty}(p_1) \subseteq T_{\infty}(p_2))=1.$
We prove that, in the random stirring model of parameter T on an infinite rooted tree each of whose vertices has at least two offspring, infinite cycles exist almost surely, provided that T is sufficiently high. In the appendices, the bound on degree above which the result holds is improved slightly.
We prove local convergence results of rerooted conditioned multi-type Galton--Watson trees. The limit objects are multitype variants of the random sin-tree constructed by Aldous (1991), and differ according to which types recur infinitely often along the backwards growing spine.
The present paper addresses the following question: for a geometric random tree in $\R^{2}$, how many semi-infinite branches cross the circle $\CC_{r}$ centered at the origin and with a large radius $r$? We develop a method ensuring that the expectation of the number $χ_{r}$ of these semi-infinite branches is $o(r)$. The result follows from the fact that, far from the origin, the distribution of the tree is close to that of an appropriate directed forest which lacks bi-infinite paths. In order to illustrate its robustness, the method is applied to three different models: the Radial Poisson Tre…
Let ${\cal G}$ be the incipient infinite cluster (IIC) for percolation on a homogeneous tree of degree $n_0+1$. We obtain estimates for the transition density of the continuous time simple random walk $Y$ on ${\cal G}$; the process satisfies anomalous diffusion and has spectral dimension 4/3.
In the case of neutral populations of fixed sizes in equilibrium whose genealogies are described by the Kingman $N$-coalescent back from time $t$ consider the associated processes of total tree length as $t$ increases. We show that the (càdlàg) process to which the sequence of compensated tree length processes converges as $N$ tends to infinity is a process of infinite quadratic variation; therefore this process cannot be a semimartingale. This answers a question posed in Pfaffelhuber et al. (2011).
We give a necessary and sufficient condition for the convergence in distribution of a conditioned Galton-Watson tree to Kesten's tree. This yields elementary proofs of Kesten's result as well as other known results on local limit of conditioned Galton-Watson trees. We then apply this condition to get new results, in the critical and sub-critical cases, on the limit in distribution of a Galton-Watson tree conditioned on having a large number of individuals with out-degree in a given set.
Continuous time branching models are used to create random fractals in a Euclidean space, whose Hausdorff dimension is controlled by an input parameter. Finite realizations are applied in modelling the set of sites visited in models of human and animal mobility.
Let $\mathcal{U}$ be the uniform spanning tree on $\mathbb{Z}^3$, whose probability law is denoted by $\mathbf{P}$. For $\mathbf{P}$-a.s. realization of $\mathcal{U}$, the recurrence of the the simple random walk on $\mathcal{U}$ is proved in [5] and it is also demonstrated in [8] that two independent simple random walks on $\mathcal{U}$ collide infinitely often. In this article, we will give a quantitative estimate on the number of collisions of two independent simple random walks on $\mathcal{U}$, which provides another proof of the infinite collision property of $\mathcal{U}$.
We study semi-infinite paths of the radial spanning tree (RST) of a Poisson point process in the plane. We first show that the expectation of the number of intersection points between semi-infinite paths and the sphere with radius $r$ grows sublinearly with $r$. Then, we prove that in each (deterministic) direction, there exists with probability one a unique semi-infinite path, framed by an infinite number of other semi-infinite paths of close asymptotic directions. The set of (random) directions in which there are more than one semi-infinite paths is dense in $[0,2π)$. It corresponds to possi…
We consider the spreading dynamics of two nested invasion clusters on an infinite tree. This model was defined as the chase-escape model by Kordzakhia and it admits a limit process, the birth-and-assassination process, previously introduced by Aldous and Krebs. On both models, we prove an asymptotic equivalent of the extinction probability near criticality. In the subcritical regime, we give a tail bound on the total progeny of the preys before extinction.
We consider the standard Abelian sandpile process on the Bethe lattice. We show the existence of the thermodynamic limit for the finite volume stationary measures and the existence of a unique infinite volume Markov process exhibiting features of self-organized criticality.
We consider stochastic processes with (or without) memory whose evolution is encoded by a finite or infinite rooted tree. The main goal is to compare the entropy rates of a given base process and a second one, to be considered as a perturbation of the former. The processes are described by probability measures on the boundary of the given tree, and by corresponding forward transition probabilities at the inner nodes. The comparison is in terms of Kullback-Leibler divergence. We elaborate and extend ideas and results of Böcherer and Amjad. Our extensions involve length functions on the edges of…
In this work, we study asymptotics of multitype Galton-Watson trees with finitely many types. We consider critical and irreducible offspring distributions such that they belong to the domain of attraction of a stable law, where the stability indices may differ. We show that after a proper rescaling, their corresponding height process converges to the continuous-time height process associated with a strictly stable spectrally positive Lévy process. This gives an analogue of a result obtained by Miermont in the case of multitype Galton-Watson trees with finite covariance matrices of the offsprin…
This paper concerns the Vertex Reinforced Jump Process (VRJP) and its representations as a Markov process in random environment. We show that all possible representations of the VRJP as a mixture of Markov processes can be expressed in a similar form, using a random potential and harmonic functions for an associated operator. This allows to show that the VRJP on $\mathbb{Z}^d$ (with certain initial conditions) has a unique representation, by proving that an associated Martin boundary is trivial. Moreover, on infinite trees, we construct a family of representations, that are all different when …
We take up the issue of deriving the limit as $n\to\infty$ of the GREM-like K process on a tree with $n$ levels. Under specific conditions on the parameters of the process, implying the martingality of a modification of the underlying clock process sequences, we obtain infinite level clock processes as nontrivial limits of the finite level clocks, and use them to construct a process on a suitable product space which is then shown to be the limit of the $n$ level K processes as $n\to\infty$. Some properties of the limiting, infinite level K process are established, like an expression for the as…
We consider critical percolation on Galton-Watson trees and prove quenched analogues of classical theorems of critical branching processes. We show that the probability critical percolation reaches depth $n$ is asymptotic to a tree-dependent constant times $n^{-1}$. Similarly, conditioned on critical percolation reaching depth $n$, the number of vertices at depth $n$ in the critical percolation cluster almost surely converges in distribution to an exponential random variable with mean depending only on the offspring distribution. The incipient infinite cluster (IIC) is constructed for a.e. Gal…
We study a spatial model of random permutations on trees with a time parameter $T>0$, a special case of which is the random stirring process. The model on trees was first analysed by Björnberg and Ueltschi[BU16], who established the existence of infinite cycles for $T$ slightly above a putatively identified critical value but left open behaviour at arbitrarily high values of $T$. We show the existence of infinite cycles for all $T$ greater than a constant, thus classifying behaviour for all values of $T$ and establishing the existence of a sharp phase transition. Numerical studies [BBBU15] of …
Since the works of Howard and Newman (2001), it is known that in straight radial rooted trees, with probability 1, infinite paths all have an asymptotic direction and each asymptotic direction is reached by (at least) an infinite path. Moreover, there exists a set of 'exceptionnal' directions reached by (at least) two infinite paths which is random, dense and only countable in dimension 2. Howard and Newman's method says nothing about (random) directions reached by more than two infinite paths and, in particular, if such 'very exceptionnal' directions exist in dimension 2. In this paper, we pr…
Random-cluster measures on infinite regular trees are studied in conjunction with a general type of `boundary condition', namely an equivalence relation on the set of infinite paths of the tree. The uniqueness and non-uniqueness of random-cluster measures are explored for certain classes of equivalence relations. In proving uniqueness, the following problem concerning branching processes is encountered and answered. Consider bond percolation on the family-tree $T$ of a branching process. What is the probability that every infinite path of $T$, beginning at its root, contains some vertex which …
The a.s. existence of a polymer probability in the infinite volume limit is readily obtained under general conditions of weak disorder from standard theory on multiplicative cascades or branching random walk. However, speculations in the case of strong disorder have been mixed. In this note existence of an infinite volume probability is established at critical strong disorder for which one has convergence in probability. Some calculations in support of a specific formula for the a.s. asymptotic variance of the polymer path under strong disorder are also provided.
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 $…
We consider a natural destruction process of an infinite recursive tree by removing each edge after an independent exponential time. The destruction up to time t is encoded by a partition $Π$(t) of N into blocks of connected vertices. Despite the lack of exchangeability, just like for an exchangeable fragmentation process, the process $Π$ is Markovian with transitions determined by a splitting rates measure r. However, somewhat surprisingly, r fails to fulfill the usual integrability condition for the dislocation measure of exchangeable fragmentations. We further observe that a time-dependent …
Combinatorial trees can be used to represent genealogies of asexual individuals. These individuals can be endowed with birth and death times, to obtain a so-called `chronological tree'. In this work, we are interested in the continuum analogue of chronological trees in the setting of real trees. This leads us to consider totally ordered and measured trees, abbreviated as TOM trees. First, we define an adequate space of TOM trees and prove that under some mild conditions, every compact TOM tree can be represented in a unique way by a so-called contour function, which is right-continuous, admits…
We show that large critical multi-type Galton-Watson trees, when conditioned to be large, converge locally in distribution to an infinite tree which is analoguous to Kesten's infinite monotype Galton-Watson tree. This is proven when we condition on the number of vertices of one fixed types, and with an extra technical assumption if we count at least two types. We then apply these results to study local limits of random planar maps, showing that large critical Boltzmann-distributed random maps converge in distribution to an infinite map.
Let $τ$n be a random tree distributed as a Galton-Watson tree with geometric offspring distribution conditioned on {Zn = an} where Zn is the size of the n-th generation and (an, n $\in$ N *) is a deterministic positive sequence. We study the local limit of these trees $τ$n as n $\rightarrow$ $\infty$ and observe three distinct regimes: if (an, n $\in$ N *) grows slowly, the limit consists in an infinite spine decorated with finite trees (which corresponds to the size-biased tree for critical or subcritical offspring distributions), in an intermediate regime, the limiting tree is composed of an…
Given i.i.d. data from an unknown distribution, we consider the problem of predicting future items. An adaptive way to estimate the probability density is to recursively subdivide the domain to an appropriate data-dependent granularity. A Bayesian would assign a data-independent prior probability to "subdivide", which leads to a prior over infinite(ly many) trees. We derive an exact, fast, and simple inference algorithm for such a prior, for the data evidence, the predictive distribution, the effective model dimension, and other quantities.
In the regime of Galton-Watson trees, first order logic statements are roughly equivalent to examining the presence of specific finite subtrees. We consider the space of all trees with Poisson offspring distribution and show that such finite subtrees will be almost surely present when the tree is infinite. Introducing the notion of universal trees, we show that all first order sentences of quantifier depth $k$ depend only on local neighbourhoods of the root of sufficiently large radius depending on $k$. We compute the probabilities of these neighbourhoods conditioned on the tree being infinite…
We study the structure of infinite geodesics in the Planar Stochastic Hyperbolic Triangulations $\mathbb{T}_λ$, which are the hyperbolic analogs of the UIPT. We prove that these geodesics form a supercritical Galton--Watson tree with geometric offspring distribution. The tree of infinite geodesics in $\mathbb{T}_λ$ provides a new notion of boundary, which is a realization of the Poisson boundary. By scaling limits arguments, we also obtain a description of the tree of infinite geodesics in the hyperbolic Brownian plane. Finally, by combining our main result with a forthcoming paper, we obtain …
In this paper we study the Maki-Thompson rumor model on infinite Cayley trees. The basic version of the model is defined by assuming that a population represented by a graph is subdivided into three classes of individuals: ignorants, spreaders and stiflers. A spreader tells the rumor to any of its (nearest) ignorant neighbors at rate one. At the same rate, a spreader becomes a stifler after a contact with other (nearest neighbor) spreaders, or stiflers. In this work we study this model on infinite Cayley trees, which is formulated as a continuous-times Markov chain, and we extend our analysis …
Given i.i.d. data from an unknown distribution, we consider the problem of predicting future items. An adaptive way to estimate the probability density is to recursively subdivide the domain to an appropriate data-dependent granularity. A Bayesian would assign a data-independent prior probability to "subdivide", which leads to a prior over infinite(ly many) trees. We derive an exact, fast, and simple inference algorithm for such a prior, for the data evidence, the predictive distribution, the effective model dimension, moments, and other quantities. We prove asymptotic convergence and consiste…
Let X and Y be infinite graphs, such that the automorphism group of X is nonamenable, and the automorphism group of Y has an infinite orbit. We prove that there is no automorphism-invariant measure on the set of spanning trees in the direct product X times Y. This implies that the minimal spanning forest corresponding to i.i.d. edge-weights in such a product, has infinitely many connected components almost surely.
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 towards the root and parks on the first available spot. It is known that the parking process on $ \mathbb{B}$ exhibits a phase transition in the sense that either a finite number of cars do not manage to park in expectation (subcritical regime) or all vertices of the tree contain a car and infinitely many cars do not manage to park (supercritical regime). We ch…
We prove the existence of a limit of the finite volume probability measures generated by tree growth rules in Ford's alpha model of phylogenetic trees. The limiting measure is shown to be concentrated on the set of trees consisting of exactly one infinite spine with finite, identically and independently distributed outgrowths.
A rotor configuration on a graph contains in every vertex an infinite ordered sequence of rotors, each is pointing to a neighbor of the vertex. After sampling a configuration according to some probability measure, a rotor walk is a deterministic process: at each step it chooses the next unused rotor in its current location, and uses it to jump to the neighboring vertex to which it points. Rotor walks capture many aspects of the expected behavior of simple random walks. However, this similarity breaks down for the property of having an infinite excursion. In this paper we study that question fo…
Consider a uniformly sampled random $d$-regular graph on $n$ vertices. If $d$ is fixed and $n$ goes to $\infty$ then we can relate typical (large probability) properties of such random graph to a family of invariant random processes (called "typical" processes) on the infinite $d$-regular tree $T_d$. This correspondence between ergodic theory on $T_d$ and random regular graphs is already proven to be fruitful in both directions. This paper continues the investigation of typical processes with a special emphasis on entropy. We study a natural notion of micro-state entropy for invariant processe…
We study invasion percolation on Aldous' Poisson-weighted infinite tree, and derive two distinct Markovian representations of the resulting process. One of these is the $σ\to\infty$ limit of a representation discovered by Angel et al. [Ann. Appl. Probab. 36 (2008) 420-466]. We also introduce an exploration process of a randomly weighted Poisson incipient infinite cluster. The dynamics of the new process are much more straightforward to describe than those of invasion percolation, but it turns out that the two processes have extremely similar behavior. Finally, we introduce two new "stationary"…
We study a class of Hermitian random matrices which includes and generalizes Wigner matrices, heavy-tailed random matrices, and sparse random matrices such as the adjacency matrices of Erdos-Renyi random graphs with p ~ 1/N. Our NxN random matrices have real entries which are i.i.d. up to symmetry. The distributions may depend on N, however, the sums of rows must converge in distribution; it is then well-known that the limiting distributions are infinitely divisible. We show that a limiting empirical spectral distribution (LSD) exists, and via local weak convergence of associated graphs, the L…
Using generating functions techniques we develop a relation between the Hausdorff and spectral dimension of trees with a unique infinite spine. Furthermore, it is shown that if the outgrowths along the spine are independent and identically distributed, then both the Hausdorff and spectral dimension can easily be determined from the probability generating function of the random variable describing the size of the outgrowths at a given vertex, provided that the probability of the height of the outgrowths exceeding n falls off as the inverse of n. We apply this new method to both critical non-gen…
