Searcharxiv⌕ Search

arXiv subjects

Ecaterina Sava-Huss

Publications and source records attributed to Ecaterina Sava-Huss.

At least 19 recordsLinked to original sources
arXiv2026

Limit shape, avalanches, and stabilisation of abelian sandpiles on comb lattices

We study two aspects of the abelian sandpile model on comb lattices. First, we prove that the infinite-volume limit of the stationary measures is supported on the saturated configuration. We then investigate the shape of avalanches induced by adding a particle at the origin in finite boxes of size $(2n+1)\times(2n+1)$ around the origin. In the stationary distribution on these finite boxes, we show that avalanches reach the boundary along the vertical teeth with probability tending to $1$, while the horizontal spread is of order $\sqrt{n}$. Finally, we establish that the single-source limit shape for abelian sandpiles on the comb lattice agrees with the corresponding limit shapes for the divisible sandpile, internal diffusion-limited aggregation (IDLA), and rotor-router aggregation models. This establishes limit shape universality on the comb lattice.

math.PR↗
arXiv2026

Phase transition for the asymptotic entropy of branching random walks on groups

We consider supercritical branching random walks (BRW) on countable groups $G$ and we prove that the asymptotic entropy of the empirical distributions of the BRW has a phase transition at $ρ_* = e^{h(μ)}$, where $h(μ)$ is the asymptotic entropy of the underlying random walk on $G$ with step distribution $μ$. Below this value $ρ_*$, the asymptotic empirical entropy of BRW equals the logarithm of the exponential growth rate of the population. Above this value, it is constantly equal to the asymptotic entropy of the underlying random walk. In particular, this answers questions from Kaimanovich-Woess [MR4663513, Section 6.3] about the existence and the behavior of the asymptotic entropy.

math.PR↗
arXiv2026

Maximal and minimal displacement of supercritical branching random walks on free products of groups

We prove that the maximal and minimal displacement of branching random walks with mean offspring number $ρ>1$ on free products of finite groups grows linearly almost surely. More precisely, we establish that the linear speed for the maximal (respectively minimal) displacement is given by the largest (respectively smallest) intersection point of the large deviation rate function of the underlying random walk with the horizontal line at height $\logρ$. The proof is based on constructing an associated multitype branching process which consists of particles that travel fast enough, and distinguishing the types via the suffix of the particles locations.

math.PR↗
arXiv2026

On the structure of the sandpile identity element on Sierpinski gasket graphs

We consider the identity of the abelian sandpile group of finite approximation graphs of the Sierpinski gasket, and we show that the second-order term in the scaling limit converges to the path distance to the nearest corner on the Sierpinski gasket. The proof relies on a decomposition of the identity of the sandpile group into the sum of a constant function and the Laplacian of the graph distance on the approximating graphs.

math.CO↗
arXiv2026

Divisible sandpiles via random walks in random scenery

We analyze an optimal stopping problem for random walk in random scenery on general graphs, and determine when it has a finite optimum. We use this to extend a theorem of Levine, Murugan, Peres, and Ugurcan [2016]. They proved that on a vertex-transitive graph, the divisible sandpile with i.i.d. initial masses of mean $μ$ stabilizes almost surely if $μ< 1$, explodes if $μ> 1$, and explodes if $μ= 1$ with positive finite variance. Their proofs rely on conservation of mean mass under toppling. This conservation extends to unimodular random graphs, but fails on general graphs. We prove explosion for all infinite bounded-degree graphs whenever $μ\geq 1$, and stabilization for $μ<1$ provided the initial masses have finite $p$-th moment for some $p>3$. Our conditions are nearly sharp: we exhibit unbounded-degree graphs on which sandpiles with $μ> 1$ stabilize, and for every $p < 3$ we construct bounded-degree graphs on which sandpiles with~$μ< 1$ and finite $p$-th moment explode.

math.PR↗
arXiv2026

Stabilization of stochastic networks in Markovian environment

We establish criteria under which stochastic networks in a Markovian environment stabilize, thus confirming Conjecture 7.2 from Levine-Greco [GL23]. The networks evolve on finite connected graphs $G=(V,E)$, and their dynamics are encoded by $V \times V$ toppling matrices $M$, whose columns record the expected number of topplings when the environment is in stationarity. Stabilization and non-stabilization are characterized by a parameter $ρ$ which depends on the largest eigenvalue of the matrix $M+αI$, with $α=1+\max\{-M(v,v):v\in V\}$. The proofs rely on the toppling random walk, in which toppled vertices are sampled according to the eigenvector associated with the largest eigenvalue of $M$.

math.PR↗
arXiv2025

Locally Markov walks on finite graphs

Locally Markov walks are natural generalizations of classical Markov chains, where instead of a particle moving independently of the past, it decides where to move next depending on the last action performed at the current location. We introduce the concept of locally Markov walks and we describe their stationary distribution and recurrent states, and we prove several properties such as irreducibility and ergodicity. For a particular locally Markov walk - the uniform unicycle walk on the complete graph - we investigate the mixing time and we prove that it exhibits cutoff.

math.PR↗
arXiv2025

Average height for Abelian sandpiles and the looping constant on Sierpinski graphs

For the Abelian sandpile model on Sierpinski graphs, we investigate several statistics such as average height, height probabilities and looping constant. In particular, we calculate the expected average height of a recurrent sandpile on the finite iterations of the Sierpinski gasket and we also give an algorithmic approach for calculating the height probabilities of recurrent sandpiles under stationarity 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 $i\in\{0,1,2,3\}$ of sandpiles under stationarity and relate the bulk average height to the looping constant on the Sierpinski gasket.

math.PR↗
arXiv2024

Sandpiles on the Vicsek fractal explode with probability 1/4

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.

math.PR↗
arXiv2024

Gaussian fluctuations for the two urn model

We introduce a modification of the generalized Pólya 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 $\sqrtρ$ and not at $ρ/2$, where $ρ$ is the dominant eigenvalue of the mean replacement matrix.

math.PR↗
arXiv2024

Random rotor walks and i.i.d. sandpiles on Sierpinski graphs

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.

math.PR↗
arXiv2023

Scaling limit of the sandpile identity element on the Sierpinski gasket

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.

math.CO↗
arXiv2023

Internal aggregation models with multiple sources and obstacle problems on Sierpinski gaskets

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 $σ_n$ 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 $σ:SG\to\mathbb{N}$. For $σ$ 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 $σ_n$ on the approximating graphs $SG_n$ converge pointwise to the continuous function $σ$ 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 $σ_n$ 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.

math.PR↗
arXiv2023

Limit theorems for discrete multitype branching processes counted with a characteristic

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 $(Z_n^Φ)_{n\in\mathbb{N}}$ such that, for each $n\in \mathbb{N}$, the contribution of each individual $u\in\mathbb{T}$ to $Z_n^Φ$ is determined by a (random) characteristic $Φ$ evaluated at the age of $u$ at time $n$. In other words, $Z_n^Φ$ is obtained by summing over all $u\in \mathbb{T}$ the corresponding contributions $Φ_u$, where $(Φ_u)_{u\in \mathbb{T}}$ are i.i.d. copies of $Φ$. Such processes are known in the literature under the name of Crump-Mode-Jagers (CMJ) processes counted with characteristic $Φ$. We derive a LLN and a CLT for the process $(Z_n^Φ)_{n\in\mathbb{N}}$ 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].

math.PR↗
arXiv2022

Abelian sandpiles on Sierpinski gasket graphs

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.

math.CO↗
arXiv2022

An epidemic model in inhomogeneous environment

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.

math.PR↗
arXiv2020

Internal DLA on Sierpinski gasket graphs

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.

math.PR↗