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 $h_* > 0$.
We consider the bond percolation problem on a transient weighted graph induced by the excursion sets of the Gaussian free field on the corresponding cable system. Owing to the continuity of this setup and the strong Markov property of the field on the one hand, and the links with potential theory for the associated diffusion on the other, we rigorously determine the behavior of various key quantities related to the (near-)critical regime for this model. In particular, our results apply in case the base graph is the three-dimensional cubic lattice. They unveil the values of the associated criti…
We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on a version of the triangular lattice in the complex plane have unique scaling limits, which are invariant under rotations, scalings, and, in the case of the MST, also under translations. However, they are not expected to be conformally invariant. We also prove some geometric properties of the limiting MST. The topology of convergence is the space of spanning trees introduced by Aizenman, Burchard, Newman & Wilson (1999), and the proof relies on the existence and conformal covariance of the scaling limit of the near-cri…
The minimal spanning forest on $\mathbb{Z}^{d}$ is known to consist of a single tree for $d \leq 2$ and is conjectured to consist of infinitely many trees for large $d$. In this paper, we prove that there is a single tree for quasi-planar graphs such as $\mathbb{Z}^{2}\times {\{0,\ldots,k\}}^{d-2}$. Our method relies on generalizations of the "Gluing Lemma" of arXiv:1401.7130. A related result is that critical Bernoulli percolation on a slab satisfies the box-crossing property. Its proof is based on a new Russo-Seymour-Welsh type theorem for quasi-planar graphs. Thus, at criticality, the proba…
Recently, Holmes and Perkins identified conditions which ensure that for a class of critical lattice models the scaling limit of the range is the range of super-Brownian motion. One of their conditions is an estimate on a spatial moment of order higher than four, which they verified for the sixth moment for spread-out lattice trees in dimensions $d>8$. Chen and Sakai have proved the required moment estimate for spread-out critical oriented percolation in dimensions $d+1>4+1$. We prove estimates on all moments for the spread-out critical contact process in dimensions $d>4$, which in particular …
We consider invasion percolation on a rooted regular tree. For the infinite cluster invaded from the root, we identify the scaling behavior of its $r$-point function for any $r\geq2$ and of its volume both at a given height and below a given height. We find that while the power laws of the scaling are the same as for the incipient infinite cluster for ordinary percolation, the scaling functions differ. Thus, somewhat surprisingly, the two clusters behave differently; in fact, we prove that their laws are mutually singular. In addition, we derive scaling estimates for simple random walk on the …
For horocyclic products of percolation subtrees of regular trees, we show almost sure amenability. Under a symmetry condition concerning the growth of the two percolation trees, we show the existence of an increasing Foelner sequence (which we call strong amenability).
We consider Bernoulli (bond) percolation with parameter $p$ on the Cayley tree of order $k$. We introduce the notion of zebra-percolation that is percolation by paths of alternating open and closed edges. In contrast with standard percolation with critical threshold at $p_c= 1/k$, we show that zebra-percolation occurs between two critical values $p_{{\rm c},1}$ and $p_{{\rm c},2}$ (explicitly given). We provide the specific formula of zebra-percolation function.
This paper surveys the results of recent collaborations with Eric Derbez and with Takashi Hara, which show that intergrated super-Brownian excursion (ISE) arises as the scaling limit of both lattice trees and the incipient infinite percolation cluster, in high dimensions. A potential extension to oriented percolation is also mentioned.
We study Bernoulli bond percolation on a random recursive tree of size $n$ with percolation parameter $p(n)$ converging to $1$ as $n$ tends to infinity. The sizes of the percolation clusters are naturally stored in a tree. We prove convergence in distribution of this tree to the genealogical tree of a continuous-state branching process in discrete time. As a corollary we obtain the asymptotic sizes of the largest and next largest percolation clusters, extending thereby a recent work of Bertoin (2014) which deals with cluster sizes in the supercritical regime. In a second part, we show that the…
This text is based on a lecture for the Sheffield Probability Day; its main purpose is to survey some recent asymptotic results about Bernoulli bond percolation on certain large random trees with logarithmic height. We also provide a general criterion for the existence of giant percolation clusters in large trees, which answers a question raised by David Croydon.
In this paper, we consider random trees associated with the genealogy of Crump-Mode-Jagers processes and perform Bernoulli bond-percolation whose parameter depends on the size of the tree. Our purpose is to show the existence of a giant percolation cluster for appropriate regimes as the size grows. We stress that the family trees of Crump-Mode-Jagers processes include random recursive trees, preferential attachment trees, binary search trees for which this question has been answered by Bertoin, as well as (more general) m-ary search trees, fragmentation trees, median-of-(2l+1) binary search tr…
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.
In this article, we first extend the construction of random interlacements, introduced by A.S. Sznitman in [arXiv:0704.2560], to the more general setting of transient weighted graphs. We prove the Harris-FKG inequality for this model and analyze some of its properties on specific classes of graphs. For the case of non-amenable graphs, we prove that the critical value u_* for the percolation of the vacant set is finite. We also prove that, once G satisfies the isoperimetric inequality IS_6 (see (1.5)), u_* is positive for the product GxZ (where we endow Z with unit weights). When the graph unde…
We consider the simple random walk on the infinite cluster of the Bernoulli bond percolation of trees, and investigate the relation between the speed of the simple random walk and the retaining probability p by studying three classes of trees. A sufficient condition is established for Galton-Watson trees.
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 analyze the metastable states near criticality of the bootstrap percolation on Galton-Watson trees. We find that, depending on the exact choice of the offspring distribution, it is possible to have several distinct metastable states, with varying scaling of their duration while approaching criticality.
We consider the model of random trees introduced by Devroye (1999), the so-called random split trees. The model encompasses many important randomized algorithms and data structures. We then perform supercritical Bernoulli bond-percolation on those trees and obtain a precise weak limit theorem for the sizes of the largest clusters. We also show that the approach developed in this work may be useful for studying percolation on other classes of trees with logarithmic height, for instance, we study also the case of $d$-regular trees.
The metric $D_α(q,q')$ on the set $Q$ of particle locations of a homogeneous Poisson process on $R^d$, defined as the infimum of $(\sum_i |q_i - q_{i+1}|^α)^{1/α}$ over sequences in $Q$ starting with $q$ and ending with $q'$ (where $| . |$ denotes Euclidean distance) has nontrivial geodesics when $α> 1$. The cases $1 <α< \infty$ are the Euclidean first-passage percolation (FPP) models introduced earlier by the authors while the geodesics in the case $α= \infty$ are exactly the paths from the Euclidean minimal spanning trees/forests of Aldous and Steele. We compare and contrast results and conj…
A split tree of cardinality $n$ is constructed by distributing $n$ "balls" in a subset of vertices of an infinite tree which encompasses many types of random trees such as $m$-ary search trees, quad trees, median-of-$(2k+1)$ trees, fringe-balanced trees, digital search trees and random simplex trees. In this work, we study Bernoulli bond percolation on arbitrary split trees of large but finite cardinality $n$. We show for appropriate percolation regimes that depend on the cardinality $n$ of the split tree that there exists a unique giant cluster, the fluctuations of the size of the giant clust…
We examine bootstrap percolation on a regular (b+1)-ary tree with initial law given by Bernoulli(p). The sites are updated according to the usual rule: a vacant site becomes occupied if it has at least theta occupied neighbors, occupied sites remain occupied forever. It is known that, when b>theta>1, the limiting density q=q(p) of occupied sites exhibits a jump at some p_t=p_t(b,theta) in (0,1) from q_t:=q(p_t)<1 to q(p)=1 when p>p_t. We investigate the metastable behavior associated with this transition. Explicitly, we pick p=p_t+h with h>0 and show that, as h decreases to 0, the system linge…
We consider a Bernoulli bond percolation on a random recursive tree of size $n\gg 1$, with supercritical parameter $p_n=1-c/\ln n$ for some $c>0$ fixed. It is known that with high probability, there exists then a unique giant cluster of size $G_n\sim \e^{-c}$, and it follows from a recent result of Schweinsberg \cite{Sch} that $G_n$ has non-gaussian fluctuations. We provide an explanation of this by analyzing the effect of percolation on different phases of the growth of recursive trees. This alternative approach may be useful for studying percolation on other classes of trees, such as for ins…
Häggström, Peres, and Steif (1997) have introduced a dynamical version of percolation on a graph $G$. When $G$ is a tree they derived a necessary and sufficient condition for percolation to exist at some time $t$. In the case that $G$ is a spherically symmetric tree, Häggström, Peres, and Steif (1997) derived a necessary and sufficient condition for percolation to exist at some time $t$ in a given target set $D$. The main result of the present paper is a necessary and sufficient condition for the existence of percolation, at some time $t\in D$, in the case that the underlying tree is not neces…
We consider a general enough set-up and obtain a refinement of the coupling between the Gaussian free field and random interlacements recently constructed by Titus Lupu in arXiv:1402.0298. We apply our results to level-set percolation of the Gaussian free field on a $(d+1)$-regular tree, when $d \ge 2$, and derive bounds on the critical value $h_*$. In particular, we show that $0 < h_* < \sqrt{2u_*}$, where $u_*$ denotes the critical level for the percolation of the vacant set of random interlacements on a $(d+1)$-regular tree.
We consider invasion percolation on Galton-Watson trees. On almost every Galton-Watson tree, the invasion cluster almost surely contains only one infinite path. This means that for almost every Galton-Watson tree, invasion percolation induces a probability measure on infinite paths from the root. We show that under certain conditions of the progeny distribution, this measure is absolutely continuous with respect to the limit uniform measure. This confirms that invasion percolation, an efficient self-tuning algorithm, may be used to sample approximately from the limit uniform distribution. Addi…
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…
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.
We study random walks on supercritical percolation clusters on wedges in $\Z^3$, and show that the infinite percolation cluster is (a.s.) transient whenever the wedge is transient. This solves a question raised by O. Haggstrom and E. Mossel. We also show that for convex gauge functions satisfying a mild regularity condition, the existence of a finite energy flow on $Z^2$ is equivalent to the (a.s.) existence of a finite energy flow on the supercritical percolation cluster. This solves a question of C. Hoffman.
We show that the transience or recurrence of a random walk in certain random environments on an arbitrary infinite locally finite tree is determined by the branching number of the tree, which is a measure of the average number of branches per vertex. This generalizes and unifies previous work of the authors. It also shows that the point of phase transition for edge-reinforced random walk is likewise determined by the branching number of the tree. Finally, we show that the branching number determines the rate of first-passage percolation on trees, also known as the first-birth problem. Our tech…
We study a percolation process on the planted binary tree, where clusters freeze as soon as they become larger than some fixed parameter N. We show that as N goes to infinity, the process converges in some sense to the frozen percolation process introduced by Aldous. In particular, our results show that the asymptotic behaviour differs substantially from that on the square lattice, on which a similar process has been studied recently by van den Berg, de Lima and Nolin.
We consider Bernoulli bond percolation on a large scale-free tree in the supercritical regime, meaning informally that there exists a giant cluster with high probability. We obtain a weak limit theorem for the sizes of the next largest clusters, extending a recent result for large random recursive trees. The approach relies on the analysis of the asymptotic behavior of branching processes subject to rare neutral mutations, which may be of independent interest.
We establish several equivalent characterisations of the anchored isoperimetric dimension of supercritical clusters in Bernoulli bond percolation on transitive graphs. We deduce from these characterisations together with a theorem of Duminil-Copin, Goswami, Raoufi, Severo, and Yadin that if $G$ is a transient transitive graph then the infinite clusters of Bernoulli percolation on $G$ are transient for $p$ sufficiently close to $1$. It remains open to extend this result down to the critical probability. Along the way we establish two new cluster repulsion inequalities that are of independent in…
Bootstrap percolation is a cellular automaton modelling the spread of an `infection' on a graph. In this note, we prove a family of lower bounds on the critical probability for $r$-neighbour bootstrap percolation on Galton--Watson trees in terms of moments of the offspring distributions. With this result we confirm a conjecture of Bollobás, Gunderson, Holmgren, Janson and Przykucki. We also show that these bounds are best possible up to positive constants not depending on the offspring distribution.
We show that critical percolation on a product of two regular trees of degree $\ge$ 3 satisfies the triangle condition. The proof does not examine the degrees of vertices and is not "perturbative" in any sense. It relies on an unpublished lemma of Oded Schramm.
An infinite graph G has the property that a random walk in random environment on G defined by i.i.d. resistances with any common distribution is almost surely transient, if and only if for some p<1, simple random walk is transient on a percolation cluster of G under bond percolation with parameter p.
In frozen percolation, i.i.d. uniformly distributed activation times are assigned to the edges of a graph. At its assigned time, an edge opens provided neither of its endvertices is part of an infinite open cluster; in the opposite case, it freezes. Aldous (2000) showed that such a process can be constructed on the infinite 3-regular tree and asked whether the event that a given edge freezes is a measurable function of the activation times assigned to all edges. We give a negative answer to this question, or, using an equivalent formulation and terminology introduced by Aldous and Bandyopadhya…
We explore the survival function for percolation on Galton-Watson trees. Letting $g(T,p)$ represent the probability a tree $T$ survives Bernoulli percolation with parameter $p$, we establish several results about the behavior of the random function $g(\mathbf{T} , \cdot)$, where $\mathbf{T}$ is drawn from the Galton-Watson distribution. These include almost sure smoothness in the supercritical region; an expression for the $k\text{th}$-order Taylor expansion of $g(\mathbf{T} , \cdot)$ at criticality in terms of limits of martingales defined from $\mathbf{T}$ (this requires a moment condition d…
We study a rumour model from a percolation theory and branching process point of view. The existence of a giant component is related to the event where the rumour, which started from the root of a tree, spreads out through an infinite number of its vertices. We present lower and upper bounds for the probability of that event, according to the distribution of the random variables that defines the radius of influence of each individual. We work with Galton-Watson branching trees (homogeneous and non-homogeneous) and spherically symmetric trees which includes homogeneous and $k-$periodic trees.
We prove that cluster observables of level-sets of the Gaussian free field on the hypercubic lattice $\mathbb{Z}^d$, $d\geq3$, are analytic on the whole off-critical regime $\mathbb{R}\setminus\{h_*\}$. This result concerns in particular the percolation density function $θ(h)$ and the (truncated) susceptibility $χ(h)$. As an important step towards the proof, we show the exponential decay in probability for the capacity of a finite cluster for all $h\neq h_*$, which we believe to be a result of independent interest. We also discuss the case of general transient graphs.
We consider Bernoulli bond percolation on oriented regular trees, where besides the usual short bonds, all bonds of a certain length are added. Independently, short bonds are open with probability $p$ and long bonds are open with probability $q$. We study properties of the critical curve which delimits the set of pairs $(p,q)$ for which there are almost surely no infinite paths. We also show that this curve decreases with respect to the length of the long bonds.
Consider independent bond percolation with retention probability p on a spherically symmetric tree Gamma. Write theta_Gamma(p) for the probability that the root is in an infinite open cluster, and define the critical value p_c=inf{p:theta_Gamma(p)>0}. If theta_Gamma(p_c)=0, then the root may still percolate in the corresponding dynamical percolation process at the critical value p_c, as demonstrated recently by Haggstrom, Peres and Steif. Here we relate this phenomenon to the near-critical behaviour of theta_Gamma(p) by showing that the root percolates in the dynamical percolation process if a…
Consider the class of k-independent bond, respectively site, percolations with parameter p on an infinite tree T. We derive tight bounds on p for both a.s. percolation and a.s. nonpercolation. The bounds are continuous functions of k and the branching number of T. This extends previous results by Lyons for the independent case (k=0) and by Bollobàs & Balister for 1-independent bond percolations. Central to our argumentation are moment method bounds à la Lyons supplemented by explicit percolation models à la Bollobàs & Balister. An indispensable tool is the minimality and explicit construction …
We apply the methods of algebraic reliability to the study of percolation on trees. To a complete $k$-ary tree $T_{k,n}$ of depth $n$ we assign a monomial ideal $I_{k,n}$ on $\sum_{i=1}^n k^i$ variables and $k^n$ minimal monomial generators. We give explicit recursive formulae for the Betti numbers of $I_{k,n}$ and their Hilbert series, which allow us to study explicitly percolation on $T_{k,n}$. We study bounds on this percolation and study its asymptotical behavior with the mentioned commutative algebra techniques.
We study invariant percolation processes on the d-regular tree that are obtained as a factor of an iid process. We show that the density of any factor of iid site percolation process with finite clusters is asymptotically at most (log d)/d as d tends to infinity. This bound is asymptotically optimal as it can be realized by independent sets. One implication of the result is a (1/2)-factor approximation gap, asymptotically in d, for estimating the density of maximal induced forests in locally tree-like d-regular graphs via factor of iid processes.
We consider the following, intuitively described process: at time zero, all sites of a binary tree are at rest. Each site becomes activated at a random uniform [0,1] time, independent of the other sites. As soon as a site is in an infinite cluster of activated sites, this cluster of activated sites freezes. The main question is whether a process like this exists. Aldous [Ald00] proved that this is the case for a slightly different version of frozen percolation. In this paper we construct a process that fits the intuitive description and discuss some properties.
In this note we discuss vacant set level set percolation on a transient weighted graph. It interpolates between the percolation of the vacant set of random interlacements and the level set percolation of the Gaussian free field. We employ coupling and derive a stochastic domination from which we deduce in a rather general set-up a certain monotonicity property of the percolation function. In the case of regular trees this stochastic domination leads to a strict inequality between some eigenvalues related to Ornstein-Uhlenbeck semi-groups for which we have no direct analytical proof. It underpi…
We consider supercritical Bernoulli bond percolation on a large $b$-ary tree, in the sense that with high probability, there exists a giant cluster. We show that the size of the giant cluster has non-gaussian fluctuations, which extends a result due to Schweinsberg in the case of random recursive trees. Using ideas in the recent work of Bertoin and Uribe Bravo, the approach developed in this work relies on the analysis of the sub-population with ancestral type in a system of branching processes with rare mutations, which may be of independent interest. This also allows us to establish the anal…
We study the accessibility percolation model on infinite trees. The model is defined by associating an absolute continuous random variable $X_v$ to each vertex $v$ of the tree. The main question to be considered is the existence or not of an infinite path of nearest neighbors $v_1,v_2,v_3\ldots$ such that $X_{v_1}<X_{v_2}<X_{v_3}<\cdots$ and which spans the entire graph. The event defined by the existence of such path is called {\it{percolation}}. We consider the case of the accessibility percolation model on a spherically symmetric tree with growth function given by $f(i)=\lceil (i+1)^ α\rcei…
In this paper we consider the geodesic tree in exponential last passage percolation. We show that for a large class of initial conditions around the origin, the line-to-point geodesic that terminates in a cylinder of width $o(N^{2/3})$ and length $o(N)$ agrees in the cylinder, with the stationary geodesic sharing the same end point. In the case of the point-to-point model, we consider width $δN^{2/3}$ and length up to $δ^{3/2} N/(\log(δ^{-1}))^3$ and provide lower and upper bound for the probability that the geodesics agree in that cylinder.
The fractal dimension of minimal spanning trees on percolation clusters is estimated for dimensions $d$ up to $d=5$. A robust analysis technique is developed for correlated data, as seen in such trees. This should be a robust method suitable for analyzing a wide array of randomly generated fractal structures. The trees analyzed using these techniques are built using a combination of Prim's and Kruskal's algorithms for finding minimal spanning trees. This combination reduces memory usage and allows for simulation of larger systems than would otherwise be possible. The path length fractal dimens…
