Supercritical Galton Watson Tree Research Articles

Note on the Generalized Branching Random Walk on the Galton–Watson Tree

Let ∂T be a super-critical Galton–Watson tree. Recently, the first author computed almost surely and simultaneously the Hausdorff dimensions of the sets of infinite branches of the boundary of ∂T along which the sequence SnX(t)/SnX˜(t) has a given set of limit points, where SnX(t) and SnX˜(t) are two branching random walks defined on ∂T. In this study, we are interested in the study of the speed of convergence of this sequence. More precisely, for a given sequence s=(sn), we consider Eα,s=t∈∂T:SnX(t)−αSnX˜(t)∼snasn→+∞. We will give a sufficient condition on (sn) so that Eα,s has a maximal Hausdorff and packing dimension.

Large deviations of extremes in branching random walk with regularly varying displacements

In this article, we consider a branching random walk on the real-line where displacements coming from the same parent have jointly regularly varying tails. The genealogical structure is assumed to be a supercritical Galton-Watson tree, satisfying Kesten-Stigum condition. We study the large deviations of the extremal process, formed by the appropriately normalized positions in the n-th generation and show that the large extreme-positions form clusters in the limit. As a consequence of this, we also study the large deviations of the maximum among positions at the n-th generation.

Parking on the infinite binary tree

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 (Bahl et al. in Parking on supercritical Galton–Watson trees, arXiv:1912.13062 , 2019; Goldschmidt and Przykucki in Comb Probab Comput 28:23–45, 2019) 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 characterize those regimes in terms of the law of A in an explicit way. We also study in detail the critical regime as well as the phase transition which turns out to be “discontinuous”.

Exponential rate for the contact process extinction time

We consider the extinction time of the contact process on increasing sequences of finite graphs obtained from a variety of random graph models. Under the assumption that the infection rate is above the critical value for the process on the integer line, in each case we prove that the logarithm of the extinction time divided by the size of the graph converges in probability to a (model-dependent) positive constant. The graphs we treat include various percolation models on increasing boxes of Z d or R d in their supercritical or percolative regimes (Bernoulli bond and site percolation, the occupied and vacant sets of random interlacements, excursion sets of the Gaussian free field, random geometric graphs) as well as supercritical Galton-Watson trees grown up to finite generations.

Parking on supercritical Galton-Watson tree

At each site of a supercritical Galton-Watson tree place a parking spot which can accommodate one car. Initially, an independent and identically distributed number of cars arrive at each vertex. Cars proceed towards the root in discrete time and park in the first available spot they come to. Let $X$ be the total number of cars that arrive to the root. Goldschmidt and Przykucki proved that $X$ undergoes a phase transition from being finite to infinite almost surely as the mean number of cars arriving to each vertex increases. We show that $EX$ is finite at the critical threshold, describe its growth rate above criticality, and prove that it increases as the initial car arrival distribution becomes less concentrated. For the canonical case that either 0 or 2 cars arrive at each vertex of a $d$-ary tree, we give improved bounds on the critical threshold and show that $P(X = 0)$ is discontinuous.

Hausdorff and packing dimensions of Mandelbrot measure

We develop, in the context of the boundary of a supercritical Galton–Watson tree, a uniform version of large deviation estimate on homogeneous trees to estimate almost surely and simultaneously the Hausdorff and packing dimensions of the Mandelbrot measure over a suitable set [Formula: see text]. As an application, we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of a fractal set related to covering number on the Galton–Watson tree.

On the cover time of [formula omitted]-biased walk on supercritical Galton–Watson trees

In this paper, we study the time required for a λ-biased (λ>1) walk to visit all the vertices of a supercritical Galton–Watson tree up to generation n. Inspired by the extremal landscape approach in Cortines et al. (2018) for the simple random walk on binary trees, we establish the scaling limit of the cover time in the biased setting.

Differentiability of the speed of biased random walks on Galton-Watson trees

We prove that the speed of a $\lambda$-biased random walk on a supercritical Galton-Watson tree is differentiable for $\lambda$ such that the walk is ballistic and obeys a central limit theorem, and give an expression of the derivative using a certain $2$-dimensional Gaussian random variable. The proof heavily uses the renewal structure of Galton-Watson trees that was introduced by Lyons-Pemantle-Peres.

The speed of a biased random walk on a Galton-Watson tree is analytic

We prove that the speed of a biased random walk on a supercritical Galton-Watson tree conditioned to survive is analytic within the ballistic regime. This extends the previous work [12] in which it was shown that the speed is differentiable within the range of bias for which a central limit theorem holds.

On the exact dimension of Mandelbrot measure

We develop, in the context of the boundary of a supercritical Galton–Watson tree, a uniform version of the argument used by Kahane 1987 on homogeneous trees to estimate almost surely and simultaneously the Hausdorff and packing dimensions of the Mandelbrot measure over a suitable set J . As an application, we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of the level sets Eα of infinite branches of the boundary of the tree along which the averages of the branching random walk have a given limit point.

Random walk on barely supercritical branching random walk

Let $${\mathcal {T}}$$ be a supercritical Galton–Watson tree with a bounded offspring distribution that has mean $$\mu >1$$, conditioned to survive. Let $$\varphi _{\mathcal {T}}$$ be a random embedding of $${\mathcal {T}}$$ into $${\mathbb {Z}}^d$$ according to a simple random walk step distribution. Let $${\mathcal {T}}_p$$ be percolation on $${\mathcal {T}}$$ with parameter p, and let $$p_c = \mu ^{-1}$$ be the critical percolation parameter. We consider a random walk $$(X_n)_{n \ge 1}$$ on $${\mathcal {T}}_p$$ and investigate the behavior of the embedded process $$\varphi _{{\mathcal {T}}_p}(X_n)$$ as $$n\rightarrow \infty $$ and simultaneously, $${\mathcal {T}}_p$$ becomes critical, that is, $$p=p_n\searrow p_c$$. We show that when we scale time by $$n/(p_n-p_c)^3$$ and space by $$\sqrt{(p_n-p_c)/n}$$, the process $$(\varphi _{{\mathcal {T}}_p}(X_n))_{n \ge 1}$$ converges to a d-dimensional Brownian motion. We argue that this scaling can be seen as an interpolation between the scaling of random walk on a static random tree and the anomalous scaling of processes in critical random environments.

Extremes of multitype branching random walks: heaviest tail wins

AbstractWe consider a branching random walk on a multitype (withQtypes of particles), supercritical Galton–Watson tree which satisfies the Kesten–Stigum condition. We assume that the displacements associated with the particles of typeQhave regularly varying tails of index$\alpha$, while the other types of particles have lighter tails than the particles of typeQ. In this paper we derive the weak limit of the sequence of point processes associated with the positions of the particles in thenth generation. We verify that the limiting point process is a randomly scaled scale-decorated Poisson point process using the tools developed by Bhattacharya, Hazra, and Roy (2018). As a consequence, we obtain the asymptotic distribution of the position of the rightmost particle in thenth generation.

Transfinite fractal dimension of trees and hierarchical scale-free graphs

Abstract In this article, we introduce a new concept: the transfinite fractal dimension of graph sequences motivated by the notion of fractality of complex networks proposed by Song et al. We show that the definition of fractality cannot be applied to networks with ‘tree-like’ structure and exponential growth rate of neighbourhoods. However, we show that the definition of fractal dimension could be modified in a way that takes into account the exponential growth, and with the modified definition, the fractal dimension becomes a proper parameter of graph sequences. We find that this parameter is related to the growth rate of trees. We also generalize the concept of box dimension further and introduce the transfinite Cesaro fractal dimension. Using rigorous proofs, we determine the optimal box-covering and transfinite fractal dimension of various models: the hierarchical graph sequence model introduced by Komjáthy and Simon, Song–Havlin–Makse model, spherically symmetric trees and supercritical Galton–Watson trees.

Supercritical causal maps: geodesics and simple random walk

We study the random planar maps obtained from supercritical Galton–Watson trees by adding the horizontal connections between successive vertices at each level. These are the hyperbolic analog of the maps studied by Curien, Hutchcroft and Nachmias in [15], and a natural model of random hyperbolic geometry. We first establish metric hyperbolicity properties of these maps: we show that they admit bi-infinite geodesics and satisfy a weak version of Gromov-hyperbolicity. We also study the simple random walk on these maps: we identify their Poisson boundary and, in the case where the underlying tree has no leaf, we prove that the random walk has positive speed. Some of the methods used here are robust, and allow us to obtain more general results about planar maps containing a supercritical Galton–Watson tree.

A quenched central limit theorem for biased random walks on supercritical Galton–Watson trees

AbstractIn this paper we prove a quenched functional central limit theorem for a biased random walk on a supercritical Galton–Watson tree with leaves. This extends a result of Peres and Zeitouni (2008) where the case without leaves was considered. A conjecture of Ben Arous and Fribergh (2016) suggests an upper bound on the bias which we observe to be sharp.

Level-set percolation for the Gaussian free field on a transient tree

We investigate level-set percolation of the Gaussian free field on transient trees, for instance on super-critical Galton-Watson trees conditioned on non-extinction. Recently developed Dynkin-type isomorphism theorems provide a comparison with percolation of the vacant set of random interlacements, which is more tractable in the case of trees. If $h_*$ and $u_*$ denote the respective (non-negative) critical values of level-set percolation of the Gaussian free field and of the vacant set of random interlacements, we show here that $h_* < \sqrt{2u}_*$ in a broad enough set-up, but provide an example where $0 = h_* = u_*$ occurs. We also obtain some sufficient conditions ensuring that $h_* > 0$.

Asymptotic properties of expansive Galton-Watson trees

We consider a super-critical Galton-Watson tree $\tau $ whose non-degenerate offspring distribution has finite mean. We consider the random trees $\tau _n$ distributed as $\tau $ conditioned on the $n$-th generation, $Z_n$, to be of size $a_n\in{\mathbb N} $. We identify the possible local limits of $\tau _n$ as $n$ goes to infinity according to the growth rate of $a_n$. In the low regime, the local limit $\tau ^0$ is the Kesten tree, in the moderate regime the family of local limits, $\tau ^\theta $ for $\theta \in (0, +\infty )$, is distributed as $\tau $ conditionally on $\{W=\theta \}$, where $W$ is the (non-trivial) limit of the renormalization of $Z_n$. In the high regime, we prove the local convergence towards $\tau ^\infty $ in the Harris case (finite support of the offspring distribution) and we give a conjecture for the possible limit when the offspring distribution has some exponential moments. When the offspring distribution has a fat tail, the problem is open. The proof relies on the strong ratio theorem for Galton-Watson processes. Those latter results are new in the low regime and high regime, and they can be used to complete the description of the (space-time) Martin boundary of Galton-Watson processes. Eventually, we consider the continuity in distribution of the local limits $(\tau ^\theta , \theta \in [0, \infty ])$.

Favorite sites of randomly biased walks on a supercritical Galton–Watson tree

Erdős and Révész (1984) initiated the study of favorite sites by considering the one-dimensional simple random walk. We investigate in this paper the same problem for a class of null-recurrent randomly biased walks on a supercritical Galton–Watson tree. We prove that there is some parameter κ∈(1,∞] such that the set of the favorite sites of the biased walk is almost surely bounded in the case κ∈(2,∞], tight in the case κ=2, and oscillates between a neighborhood of the root and the boundary of the range in the case κ∈(1,2). Moreover, our results yield a complete answer to the cardinality of the set of favorite sites in the case κ∈(2,∞]. The proof relies on the exploration of the Markov property of the local times process with respect to the space variable and on a precise tail estimate on the maximum of local times, using a change of measure for multi-type Galton–Watson trees.

Harmonic measure for biased random walk in a supercritical Galton–Watson tree

We consider random walks $\lambda $-biased towards the root on a Galton–Watson tree, whose offspring distribution $(p_{k})_{k\geq 1}$ is non-degenerate and has finite mean $m>1$. In the transient regime $0<\lambda <m$, the loop-erased trajectory of the biased random walk defines the $\lambda $-harmonic ray, whose law is the $\lambda $-harmonic measure on the boundary of the Galton–Watson tree. We answer a question of Lyons, Pemantle and Peres (In Classical and Modern Branching Processes (Minneapolis, MN, 1994) (1997) 223–237 Springer) by showing that the $\lambda $-harmonic measure has a.s. strictly larger Hausdorff dimension than the visibility measure, which is the harmonic measure corresponding to the simple forward random walk. We also prove that the average number of children of the vertices along the $\lambda $-harmonic ray is a.s. bounded below by $m$ and bounded above by $m^{-1}\sum k^{2}p_{k}$. Moreover, at least for $0<\lambda \leq 1$, the average number of children of the vertices along the $\lambda $-harmonic ray is a.s. strictly larger than that of the $\lambda $-biased random walk trajectory. We observe that the latter is not monotone in the bias parameter $\lambda $.

The slow regime of randomly biased walks on trees

We are interested in the randomly biased random walk on the supercritical Galton–Watson tree. Our attention is focused on a slow regime when the biased random walk $(X_{n})$ is null recurrent, making a maximal displacement of order of magnitude $(\log n)^{3}$ in the first $n$ steps. We study the localization problem of $X_{n}$ and prove that the quenched law of $X_{n}$ can be approximated by a certain invariant probability depending on $n$ and the random environment. As a consequence, we establish that upon the survival of the system, $\frac{|X_{n}|}{(\log n)^{2}}$ converges in law to some non-degenerate limit on $(0,\infty)$ whose law is explicitly computed.