We consider fixed-point equations for probability measures charging measured compact metric spaces that naturally yield continuum random trees. On the one hand, we study the existence/uniqueness of the fixed-points and the convergence of the corresponding iterative schemes. On the other hand, we study the geometric properties of the random measured real trees that are fixed-points, in particular their fractal properties. We obtain bounds on the Minkowski and Hausdorff dimension, that are proved tight in a number of applications, including the very classical continuum random tree, but also for the dual trees of random recursive triangulations of the disk introduced by Curien and Le Gall [Ann Probab, vol. 39, 2011]. The method happens to be especially efficient to treat cases for which the mass measure on the real tree induced by natural encodings only provides weak estimates on the Hausdorff dimension.
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 ${\mathbb P}(T_{\infty}(p_1) \subseteq T_{\infty}(p_2))=1.$
The Derrida-Retaux model on a geometric Galton-Watson tree
Institut für Mathematische Stochastik, Universität Münster（明斯特大学）；LAGA, Université Paris XIII（巴黎第十三大学）；Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, UPS（图卢兹大学）
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…
We explore the tree limits recently defined by Elek and Tardos. In particular, we find tree limits for many classes of random trees. We give general theorems for three classes of conditional Galton-Watson trees and simply generated trees, for split trees and generalized split trees (as defined here), and for trees defined by a continuous-time branching process. These general results include, for example, random labelled trees, ordered trees, random recursive trees, preferential attachment trees, and binary search trees.
We provide a new geometric representation of a family of fragmentation processes by nested laminations, which are compact subsets of the unit disk made of noncrossing chords. We specifically consider a fragmentation obtained by cutting a random stable tree at random points, which split the tree into smaller subtrees. When coding each of these cutpoints by a chord in the unit disk, we separate the disk into smaller connected components, corresponding to the smaller subtrees of the initial tree. This geometric point of view allows us in particular to highlight a new relation between the Aldous-P…
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…
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…
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…
Consider~\(n\) nodes~\(\{X_i\}_{1 \leq i \leq n}\) independently distributed in the unit square~\(S,\) each according to a distribution~\(f.\) Nodes~\(X_i\) and~\(X_j\) are joined by an edge if the Euclidean distance~\(d(X_i,X_j)\) is less than~\(r_n,\) the adjacency distance and the resulting random graph~\(G_n\) is called a random geometric graph~(RGG). We now assign a location dependent weight to each edge of~\(G_n\) and define~\(MST_n\) to be the sum of the weights of the minimum spanning trees of all components of~\(G_n.\) For values of~\(r_n\) above the connectivity regime, we obtain upp…
We introduce a simple technique for proving the transience of certain processes defined on the random tree $\mathcal{G}$ generated by a supercritical branching process. We prove the transience for once-reinforced random walks on $\mathcal{G}$, that is, a generalization of a result of Durrett, Kesten and Limic [Probab. Theory Related Fields 122 (2002) 567--592]. Moreover, we give a new proof for the transience of a family of biased random walks defined on $\mathcal{G}$. Other proofs of this fact can be found in [Ann. Probab. 16 (1988) 1229--1241] and [Ann. Probab. 18 (1990) 931--958] as part of…
We consider the motion of a particle on a Galton Watson tree, when the probabilities of jumping from a vertex to any one of its neighbours is determined by a random process. Given the tree, positive weights are assigned to the edges in such a way that, viewed along any line of descent, they evolve as a random process. In order to introduce our method for proving transience or recurrence, we first suppose that the weights are i.i.d., reproving a result of Lyons and Pemantle. We then extend the argument to allow a Markovian environment, and finally to a random walk on a Markovian environment tha…
We study $N$-ary non-commutative notions of independence, which are given by trees and which generalize free, Boolean, and monotone independence. For every rooted subtree $\mathcal{T}$ of the $N$-regular tree, we define the $\mathcal{T}$-free product of $N$ non-commutative probability spaces and we define the $\mathcal{T}$-free additive convolution of $N$ non-commutative laws. These $N$-ary convolution operations form a topological symmetric operad which includes the free, Boolean, monotone, and anti-monotone convolutions, as well as the orthogonal and subordination convolutions. Using the ope…
Let T be the homogeneous tree with degree and G a finitely generated group whose Cayley graph is T. The associated lamplighter group is the wreath product of the cyclic group of order r with G. For a large class of random walks on this group, we prove almost sure convergence to a natural geometric boundary. If the probability law governing the random walk has finite first moment, then the probability space formed by this geometric boundary together with the limit distribution of the random walk is proved to be maximal, that is, the Poisson boundary. We also prove that the Dirichlet problem at …
We define generic ensembles of infinite trees. These are limits as $N\to\infty$ of ensembles of finite trees of fixed size $N$, defined in terms of a set of branching weights. Among these ensembles are those supported on trees with vertices of a uniformly bounded order. The associated probability measures are supported on trees with a single spine and Hausdorff dimension $d_h =2$. Our main result is that their spectral dimension is $d_s=4/3$, and that the critical exponent of the mass, defined as the exponential decay rate of the two-point function along the spine, is 1/3.
We study the hyperbolic random geometric graph introduced in Krioukov et al. For a sequence $R_n \to \infty$, we define these graphs to have the vertex set as Poisson points distributed uniformly in balls $B(0,R_n) \subset B_d^α$, the $d$-dimensional Poincaré ball (unit d-ball with the Poincaré metric $d_α$ corresponding to negative curvature $-α^2, α> 0$) by connecting any two points within a distance $R_n$ according to the metric $d_ζ, ζ> 0$. Denoting these graphs by $HG_n(R_n ; α, ζ)$, we study asymptotic counts of copies of a fixed tree $Γ_k$ (with the ordered degree sequence $d_{(1)} \leq…
