We consider a recent model of random walk that recursively grows the network on which it evolves, namely the Tree Builder Random Walk (TBRW). We introduce a bias rho is an element of (0, infinity) towards the root, and exhibit a phase transition for transience/recurrence at a critical threshold rho c = 1+2 nu, where nu is the (possibly infinite) expected number of new leaves attached to the walker's position at each step. This generalizes previously known results, which focused on the unbiased case rho = 1. The proofs rely on a recursive analysis of the local times of the walk at each vertex of the tree, after a given number of returns to the root. We moreover characterize the strength of the transience (law of large numbers and central limit theorem with positive speed) via standard arguments, establish recurrence at rho c, and show a condensation phenomenon in the non-critical recurrent case.

We study a nearest-neighbor coloring process in which vertices are revealed in random order and inherit the color of the closest vertex revealed before them. This model is a discrete analogue of coloring processes previously studied by Preater (2009) and Aldous (2018) in Euclidean spaces. We focus here on regular trees and analyze the associated genealogy of color inheritance. In contrast with the Euclidean case, the genealogical graph on an infinite regular tree is not connected: it has infinitely many infinite one-ended components, each with a distinct asymptotic direction, while every vertex has only finitely many descendants. We also describe how this structure is modified in the presence of finitely many initial seeds. Finally, we study local limits of the coloring on finite regular trees as their height tends to infinity, for two natural seed configurations: two fixed seeds, and one blue seed at the root with red seeds at the leaves.

A fair coin is flipped n times, and two finite sequences of heads and tails (words) A and B of the same length are given. Each time the word A appears in the sequence of coin flips, Alice gets a point, and each time the word B appears, Bob gets a point. Who is more likely to win? This puzzle is a slight extension of Litt's game that recently set Twitter abuzz. We show that Litt's game is fair for any value of n and any two words that have the same auto-correlation structure by building up a bijection that exchanges Bob and Alice scores; the fact that the inter-correlation does not come into play in this case may come up as a surprise.

We consider the local limit of uniformly distributed directed animals with size n on the square lattice viewed from the root. Two constructions of the resulting uniform infinite directed animal are given: one as a heap of dominoes, constructed by letting gravity act on a right-continuous random walk, and one as a Markov process, obtained by slicing the animal horizontally. We look at geometric properties of this local limit and establish, in particular, that it consists of a single vertex at infinitely many (random) levels. The proof relies on martingales arguments and showcases the strength of this probabilistic approach.

We study a model of random partitioning by nearest-neighbor coloring from Poisson rain, introduced independently by Aldous and Preater. Given two initial points in $[0,1]^d$ respectively colored in red and blue, we let independent uniformly random points fall in $[0,1]^d$, and upon arrival, each point takes the color of the nearest point fallen so far. We prove that the colored regions converge in the Hausdorff sense towards two random closed subsets whose intersection, the frontier, has Hausdorff dimension strictly between $d-1$ and $d$, thus answering a conjecture raised by Aldous. However, several topological properties of the frontier remain elusive.

We revisit an unpublished paper of Vervoort (2002) on the once reinforced random walk, and prove that this process is recurrent on any graph of the form $\mathbb{Z}\times \Gamma$, with $\Gamma$ a finite graph, for sufficiently large reinforcement parameter. We also obtain a shape theorem for the set of visited sites, and show that the fluctuations around this shape are of polynomial order. The proof involves sharp general estimates on the time spent on subgraphs of the ambiant graph which might be of independent interest.

We revisit an unpublished paper of Vervoort (2002) on the once reinforced random walk, and prove that this process is recurrent on any graph of the form $\mathbb{Z}\times Γ$, with $Γ$ a finite graph, for sufficiently large reinforcement parameter. We also obtain a shape theorem for the set of visited sites, and show that the fluctuations around this shape are of polynomial order. The proof involves sharp general estimates on the time spent on subgraphs of the ambiant graph which might be of independent interest.

We consider a critical Bernoulli site percolation on the uniform infinite planar triangulation. We study the tail distributions of the peeling time, perimeter, and volume of the hull of a critical cluster. The exponents obtained here differs by a factor 2 from those computed previously by Angel and Curien (2015) in the case of critical site percolation on the uniform infinite half-plane triangulation.

We construct a stationary random tree, embedded in the upper half plane, with prescribed offspring distribution and whose vertices are the atoms of a unit Poisson point process. This process which we call Hammersley’s tree process extends the usual Hammersley’s line process. Just as Hammersley’s process is related to the problem of the longest increasing subsequence, this model also has a combinatorial interpretation: it counts the number of heaps (i.e. increasing trees) required to store a random permutation. This problem was initially considered by Byers et al. (ANALCO11, workshop on analytic algorithmics and combinatorics, pp 33–44, 2011) and Istrate and Bonchis (Partition into Heapable sequences, heap tableaux and a multiset extension of Hammersley’s process. Lecture notes in computer science combinatorial pattern matching, pp 261–271, 2015) in the case of regular trees. We show, in particular, that the number of heaps grows logarithmically with the size of the permutation.

We study the minimum number of heaps required to sort a random sequence using a generalization of Istrate and Bonchis's algorithm (2015). In a previous paper, the authors proved that the expected number of heaps grows logarithmically. In this note, we improve on the previous result by establishing the almost-sure and L-1 convergence.

Given a weighted graph, we introduce a partition of its vertex set such that the distance between any two clusters is bounded from below by a power of the minimum weight of both clusters. This partition is obtained by recursively merging smaller clusters and cumulating their weights. For several classical random weighted graphs, we show that there exists a phase transition regarding the existence of an infinite cluster. The motivation for introducing this partition arises from a connection with the contact process as it roughly describes the geometry of the sets where the process survives for a long time. We give a sufficient condition on a graph to ensure that the contact process has a non trivial phase transition in terms of the existence of an infinite cluster. As an application, we prove that the contact process admits a sub-critical phase on d-dimensional random geometric graphs and on random Delaunay triangulations. To the best of our knowledge, these are the first examples of graphs with unbounded degrees where the critical parameter is shown to be strictly positive.

We consider a vertex reinforced random walk on the integer lattice with sub-linear reinforcement. Under some assumptions on the regular variation of the weight function, we characterize whether the walk gets stuck on a finite interval. When this happens, we estimate the size of the localization set. In particular, we show that, for any odd number N larger than or equal to 5 , there exists a vertex reinforced random walk which localizes with positive probability on exactly N consecutive sites.

We characterize non-decreasing weight functions for which the associated one-dimensional vertex reinforced random walk (VRRW) localizes on 4 sites. A phase transition appears for weights of order $n\log \log n$: for weights growing faster than this rate, the VRRW localizes almost surely on at most 4 sites whereas for weights growing slower, the VRRW cannot localize on less than 5 sites. When $w$ is of order $n\log \log n$, the VRRW localizes almost surely on either 4 or 5 sites, both events happening with positive probability.

We prove that any vertex-reinforced random walk on the integer lattice with non-decreasing reinforcement sequence w satisfying w(k) = o(k^α) for some α < 1/2 is recurrent. This improves on previous results of Volkov (2006) and Schapira (2012).

We consider a continuous-time vertex reinforced jump process on a supercritical Galton-Watson tree. This process takes values in the set of vertices of the tree and jumps to a neighboring vertex with rate proportional to the local time at that vertex plus a constant $c$. The walk is either transient or recurrent depending on this parameter $c$. In this paper, we complete results previously obtained by Davis and Volkov [Probab. Theory Related Fields 123 (2002) 281-300, Probab. Theory Related Fields 128 (2004) 42-62] and Collevecchio [Ann. Probab. 34 (2006) 870-878, Electron. J. Probab. 14 (2009) 1936-1962] by proving that there is a unique (explicit) positive $c_{\mathrm{crit}}$ such that the walk is recurrent for $c\leq c_{\mathrm{crit}}$ and transient for $c>c_{\mathrm{crit}}$.

We study a model of multi-excited random walk on a regular tree which generalizes the models of the once excited random walk and the digging random walk introduced by Volkov (2003). We show the existence of a phase transition and provide a criterion for the recurrence/transience property of the walk. In particular, we prove that the asymptotic behaviour of the walk depends on the order of the excitations, which contrasts with the one dimensional setting studied by Zerner (2005). We also consider the limiting speed of the walk in the transient regime and conjecture that it is not a monotonic function of the environment.

We consider the model of the one-dimensional cookie random walk when the initial cookie distribution is spatially uniform and the number of cookies per site is finite. We give a criterion to decide whether the limiting speed of the walk is non-zero. In particular, we show that a positive speed may be obtained for just three cookies per site. We also prove a result on the continuity of the speed with respect to the initial cookie distribution.

We consider a one-dimensional transient cookie random walk. It is known from a previous paper that a cookie random walk $(X_n)$ has positive or zero speed according to some positive parameter $\alpha >1$ or $\le 1$. In this article, we give the exact rate of growth of $(X_n)$ in the zero speed regime, namely: for $0<\alpha <1$, $X_n/n^{\frac{\alpha+1}{2}}$ converges in law to a Mittag-Leffler distribution whereas for $\alpha=1$, $X_n(\log n)/n$ converges in probability to some positive constant.