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Vicsek fractal graphs are an important class of infinite graphs with self similar properties, polynomial growth and treelike features, on which several dynamical processes such as random walks or Abelian sandpiles can be rigorously analyzed and one can obtain explicit closed form expressions. While such processes on Vicsek fractals and on Euclidean lattices $\mathbb{Z}^2$ share some properties for instance in the recurrence behaviour, many quantities related to sandpiles on Euclidean lattices are still poorly understood. The current work focuses on the stabilization and explosion of Abelian sandpiles on Vicsek fractal graphs, and we prove that a sandpile sampled from the infinite volume limit plus one additional particle stabilizes with probability 3/4, that is, it does not stabilize almost surely and it explodes with the complementary probability 1/4. We prove the main result by using two different approaches: one of probabilistic nature and one of algebraic flavor. The first approach is based on investigating the particles sent to the boundary of finite volumes and showing that their number stays above four with positive probability. In the second approach we relate the question of stabilization and explosion of sandpiles in infinite volume to the order of elements of the sandpile group on finite approximations of the infinite Vicsek graph. The method applies to more general state spaces and by employing it we also find all invariant factors of the sandpile groups on the finite approximations of the infinite Vicsek fractal., Comment: some improvements and minor corrections!

We investigate the identity element of the sandpile group on finite approximations of the Sierpinski gasket with normal boundary conditions and show that the sequence of piecewise constant continuations of the identity elements on SG_n converges in the weak* sense to the constant function with value 4 on the Sierpinski gasket SG. We then generalize the proof to a wider range of functions and obtain the scaling limit for the identity elements with different choices of sink vertices., Comment: 13 pages, 10 figures; to appear in the proceedings Volume

We investigate the Abelian sandpile model and related quantities such as height probabilities and looping constant on fractal graphs. In particular, we give an algorithmic approach for calculating the height probabilities of recurrent sandpiles under stationarity on finite iterations of the Sierpinski graphs by using the connection between recurrent configurations of the Abelian sandpile Markov chain and uniform spanning trees. We also calculate the expected fraction of vertices of height i for $i\in\{0,1,2,3\}$ and expected bulk average height of sandpiles under stationarity and relate it to the looping constant on the Sierpinski gasket.

We introduce a modification of the generalized P\'olya urn model containing two urns and we study the number of balls $B_j(n)$ of a given color $j\in\{1,\ldots,J\}$, $J\in\mathbb{N}$ added to the urns after $n$ draws. We provide sufficient conditions under which the random variables $(B_j(n))_{n\in\mathbb{N}}$ properly normalized and centered converge weakly to a limiting random variable. The result reveals a similar trichotomy as in the classical case with one urn, one of the main differences being that in the scaling we encounter 1-periodic continuous functions. Another difference in our results compared to the classical urn models is that the phase transition of the second order behavior occurs at $\sqrt{\rho}$ and not at $\rho/2$, where $\rho$ is the dominant eigenvalue of the mean replacement matrix., Comment: to appear in Advances in Applied Probability (2024)

We consider the doubly infinite Sierpinski gasket graph $SG_0$, rescale it by factor $2^{-n}$, and on the rescaled graphs $SG_n=2^{-n}SG_0$, for every $n\in \mathbb{N}$, we investigate the limit shape of three aggregation models with initial configuration $\sigma_n$ of particles supported on multiple vertices. The models under consideration are: divisible sandpile in which the excess mass is distributed among the vertices until each vertex is stable and has mass less or equal to one, internal DLA in which particles do random walks until finding an empty site, and rotor aggregation in which particles perform deterministic counterparts of random walks until finding an empty site. We denote by $SG=cl(\cup_{n=0}^{\infty} SG_n)$ the infinite Sierpinski gasket, which is a closed subset of $\mathbb{R}^2$, for which $SG_n$ represents the level-n approximating graph, and we consider a continuous function $\sigma:SG\to\mathbb{N}$. For $\sigma$ we solve the obstacle problem and we describe the noncoincidence set $D\subset SG$ as the solution of a free boundary problem on the fractal $SG$. If the discrete particle configurations $\sigma_n$ on the approximating graphs $SG_n$ converge pointwise to the continuous function $\sigma$ on the limit set $SG$, we prove that, as $n\to\infty$, the scaling limits of the three aforementioned models on $SG_n$ starting with initial particle configuration $\sigma_n$ converge to the deterministic solution $D$ of the free boundary problem on the limit set $SG\subset\mathbb{R}^2$. For $D$ we also investigate boundary regularity properties., Comment: to appear in Journal of Fractal Geometry (2023)

We prove that, on the infinite Sierpinski gasket graph SG, rotor walk with random initial configuration of rotors is recurrent. We also give a necessary condition for an i.i.d. sandpile to stabilize. In particular, we prove that an i.i.d. sandpile with expected number of chips per site greater or equal to three does not stabilize almost surely. Furthermore, the proof also applies to divisible sandpiles and shows that divisible sandpile at critical density one does not stabilize almost surely on SG., Comment: 13 pages, 3 figures; referee's suggestions implemented; to appear in Statistics and Probability Letters (2024)

The aim of the current work is to investigate structural properties of the sandpile group of a special class of self-similar graphs. More precisely, we consider Abelian sandpiles on Sierpinski gasket graphs and for the choice of normal boundary conditions, we give a characterization of the identity element and a recursive description of the sandpile group. Finally, we consider Abelian sandpile Markov chains on the aforementioned graphs and we improve the existing bounds on the speed of convergence to stationarity.

For a discrete time multitype supercritical Galton-Watson process $(Z_n)_{n\in \mathbb{N}}$ and corresponding genealogical tree $\mathbb{T}$, we associate a new discrete time process $(Z_n^{\Phi})_{n\in\mathbb{N}}$ such that, for each $n\in \mathbb{N}$, the contribution of each individual $u\in\mathbb{T}$ to $Z_n^{\Phi}$ is determined by a (random) characteristic $\Phi$ evaluated at the age of $u$ at time $n$. In other words, $Z_n^{\Phi}$ is obtained by summing over all $u\in \mathbb{T}$ the corresponding contributions $\Phi_u$, where $(\Phi_u)_{u\in \mathbb{T}}$ are i.i.d. copies of $\Phi$. Such processes are known in the literature under the name of Crump-Mode-Jagers (CMJ) processes counted with characteristic $\Phi$. We derive a LLN and a CLT for the process $(Z_n^{\Phi})_{n\in\mathbb{N}}$ in the discrete time setting, and in particular, we show a dichotomy in its limit behavior. By applying our main result, we also obtain a generalization of the results in Kesten-Stigum [17]., Comment: a revisited version, which incorporates the referee's suggestions, and several other improvements and explanations. In particular, the section explaining how to recover the results from Kesten-Stigum [17] has been rewritten

The current work deals with an epidemic model on the complete graph K_n on n vertices in a non-homogeneous setting, where the vertices may have distinct types. Different types differ in the probability of getting infected, and/or in the capacity of infecting other vertices. This generalizes previous models where vertices are all of the same type and have equal probabilities of being infected. We prove laws of large numbers and central limit theorems for the the total duration of the process and for the number of infected vertices, respectively, when n goes to infinity. By coupling the epidemic model with a Poisson process, we also obtain continuous-time counterparts of the above-mentioned limit results. Moreover, we also prove that when all individuals have the same spread capacity, then a population with inhomogeneous susceptibility is less affected by the epidemics than a homogeneous population., Comment: final version incorporating referee's suggestions

This paper studies the boundary behaviour of $\lambda$-polyharmonic functions for the simple random walk operator on a regular tree, where $\lambda$ is complex and $|\lambda|> \rho$, the $\ell^2$-spectral radius of the random walk. In particular, subject to normalisation by spherical, resp. polyspherical functions, Dirichlet and Riquier problems at infinity are solved and a non-tangential Fatou theorem is proved.

We present an unified approach on the behavior of two random growth models (external DLA and internal DLA) on infinite graphs, the second one being an internal counterpart of the first one. Even though the two models look pretty similar, their behavior is completely different: while external DLA tends to build irregularities and fractal-like structures, internal DLA tends to fill up gaps and to produce regular clusters. We will also consider the aforementioned models on fractal graphs like Sierpinski gasket and carpet, and present some recent results and possible questions to investigate., Comment: references added; minor reviews, with referee's suggestions incorporated

The aim of the current work is to prove a law of large numbers for the range size of recurrent rotor walks with random initial configuration on a general class of trees, called periodic trees or directed covers of graphs., Comment: New references added; 21 pages, 1 figure. Appendix on rotor-recurrent trees that contain rotor-transient subtrees. arXiv admin note: text overlap with arXiv:1203.1477

We prove a law of large numbers for the range of rotor walks with random initial configuration on regular trees and on Galton-Watson trees. More precisely, we show that on the classes of trees under consideration, even in the case when the rotor walk is recurrent, the range grows at linear speed. We also show the existence of the speed for such rotor walks., Comment: Final version; accepted in Journal of Theoretical Probability 2019

The divisible sandpile model is a growth model on graphs that was introduced by Levine and Peres as a tool to study internal diffusion limited aggregation. In this work we investigate the shape of the divisible sandpile model on the graphical Sierpinski gasket SG. We show that the shape is a ball in the graph metric of SG. Moreover we give an exact representation of the odometer function of the divisible sandpile., Comment: 20 pages, 7 figures

Internal diffusion-limited aggregation (IDLA) is a stochastic growth model on a graph $G$ which describes the formation of a random set of vertices growing from the origin (some fixed vertex) of $G$. Particles start at the origin and perform simple random walks; each particle moves until it lands on a site which was not previously visited by other particles. This random set of occupied sites in $G$ is called the IDLA cluster. In this paper we consider IDLA on Sierpinski gasket graphs, and show that the IDLA cluster fills balls (in the graph metric) with probability 1., Comment: 24 pages, 2 figures. Final version, to appear as Chapter 7 of "Analysis and Geometry on Graphs and Manifolds," M. Keller, D. Lenz, and R.K. Wojciechowski, Cambridge University Press (2020)

We introduce a family of stochastic processes on the integers, depending on a parameter $p \in [0,1]$ and interpolating between the deterministic rotor walk (p=0) and the simple random walk (p=1/2). This p-rotor walk is not a Markov chain but it has a local Markov property: for each $x \in \mathbb{Z}$ the sequence of successive exits from $x$ is a Markov chain. The main result of this paper identifies the scaling limit of the p-rotor walk with two-sided i.i.d. initial rotors. The limiting process takes the form $\sqrt{\frac{1-p}{p}} X(t)$, where $X$ is a doubly perturbed Brownian motion, that is, it satisfies the implicit equation \begin{equation} X(t) = \mathcal{B}(t) + a \sup_{s\leq t} X(s) + b \inf_{s\leq t} X(s) \end{equation} for all $t \in [0,\infty)$. Here $\mathcal{B}(t)$ is a standard Brownian motion and $a,b<1$ are constants depending on the marginals of the initial rotors on $\mathbb{N}$ and $-\mathbb{N}$ respectively. Chaumont and Doney [CD99] have shown that the above equation has a pathwise unique solution $X(t)$, and that the solution is almost surely continuous and adapted to the natural filtration of the Brownian motion. Moreover, $\limsup X(t) = +\infty$ and $\liminf X(t) = -\infty$ [CDH00]. This last result, together with the main result of this paper, implies that the p-rotor walk is recurrent for any two-sided i.i.d. initial rotors and any $0

We consider random walks on non-amenable Baumslag-Solitar groups BS(p,q) and describe their Poisson-Furstenberg boundary. The latter is a probabilistic model for the long-time behaviour of the random walk. In our situation, we identify it in terms of the space of ends of the Bass-Serre tree and the real line using Kaimanovich's strip criterion., Comment: 24 pages, 7 figures, v3: revision implementing the suggestions of the reviewer

A rotor-router walk on a graph is a deterministic process, in which each vertex is endowed with a rotor that points to one of the neighbors. A particle located at some vertex first rotates the rotor in a prescribed order, and then it is routed to the neighbor the rotor is now pointing at. In the current work we make a step toward in understanding the behavior of rotor-router walks on random trees. More precisely, we consider random i.i.d. initial configurations of rotors on Galton-Watson trees, i.e. on a family tree arising from a Galton-Watson process, and give a classification in recurrence and transience for rotor-router walks on these trees., Comment: 14 pages; revised version

In this paper we investigate the connectedness and the isomorphism problems for zig-zag products of two graphs. A sufficient condition for the zig-zag product of two graphs to be connected is provided, reducing to the study of the connectedness property of a new graph which depends only on the second factor of the graph product. We show that, when the second factor is a cycle graph, the study of the isomorphism problem for the zig-zag product is equivalent to the study of the same problem for the associated pseudo-replacement graph. The latter is defined in a natural way, by a construction generalizing the classical replacement product, and its degree is smaller than the degree of the zig-zag product graph. Two particular classes of products are studied in detail: the zig-zag product of a complete graph with a cycle graph, and the zig-zag product of a $4$-regular graph with the cycle graph of length $4$. Furthermore, an example coming from the theory of Schreier graphs associated with the action of self-similar groups is also considered: the graph products are completely determined and their spectral analysis is developed., Comment: 33 pages

The aim of this note is to extend the result of Angel and Holroyd concerning the transience and the recurrence of transfinite rotor-router walks, for random initial configuration of rotors on homogeneous trees. We address the same question on directed covers of finite graphs, which are also called trees with finitely many cone types or periodic trees. Furthermore, we provide an example of a directed cover such that the rotor-router walk can be either recurrent or transient, depending only on the planar embedding of the periodic tree., Comment: 14 pages, 3 figures; The previous version contains an error in the proof of Corollary 3.8, which is essential for the transience part of the main result. This is now corrected by constructing a new rotor-router process, which we call the frontier process