arXiv subjects

Giulio Iacobelli

Publications and source records attributed to Giulio Iacobelli.

17 recordsLinked to original sources

arXiv2026

Generalized Excited Random Walks under Bernoulli excitations

We study a variant of the Generalized Excited Random Walk (GERW) on $\mathbb{Z}^d$ introduced by Menshikov, Popov, Ramírez and Vachkovskaia in [Ann. Probab. 40 (5), 2012]. It consists of a particular version of the model studied in [arXiv preprint arXiv:2211.05715, 2022] where excitation may or may not occur according to a time-dependent probability. Specifically, given $\{p_n\}_{n \ge 1}$, $p_n \in (0, 1]$ for all $n \ge 1$, whenever the process visits a site at time $n$ for the first time, with probability $p_n$ it gains a drift in a fixed direction. Otherwise, it behaves as a $d$-martingale with zero-mean vector. We refer to the model as $p_n$-GERW. Assuming bounded jumps and $p_n \approx n^{-β}$, we show a series of results for the $p_n$-\Name{} depending on the value of $β$ and on the dimension $d$. Specifically, for every $β\in(0,1]$ and $d=2$ or $d>h(β)$, with $h$ a decreasing function of $β$, we prove a SLLN for the range, while for $β<1/2$ we prove a sub-ballistic SLLN for the process whenever the SLLN for the range holds. We also study the $p_n$-\Name{} under diffusive scaling, and we obtain a Functional Central Limit Theorem for $β> 1/2$ and $d\geq 2$, or $β=1/2$ and $d=2$. Finally, for $β=1/2$ and $d \ge 11$ we show that the diffusively rescaled $p_n$-\Name{} converges in distribution to a Brownian Motion plus a multiple of the square root of time.

math.PR↗

arXiv2025

hdMTD: An R Package for High-Dimensional Mixture Transition Distribution Models

Several natural phenomena exhibit long-range conditional dependencies. High-order mixture transition distribution (MTD) are parsimonious non-parametric models to study these phenomena. An MTD is a Markov chain in which the transition probabilities are expressed as a convex combination of lower-order conditional distributions. Despite their generality, inference for MTD models has traditionally been limited by the need to estimate high-dimensional joint distributions. In particular, for a sample of size n, the feasible order d of the MTD is typically restricted to d approximately O(log n). To overcome this limitation, Ost and Takahashi (2023) recently introduced a computationally efficient non-parametric inference method that identifies the relevant lags in high-order MTD models, even when d is approximately O(n), provided that the set of relevant lags is sparse. In this article, we introduce hdMTD, an R package allowing us to estimate parameters of such high-dimensional Markovian models. Given a sample from an MTD chain, hdMTD can retrieve the relevant past set using the BIC algorithm or the forward stepwise and cut algorithm described in Ost and Takahashi (2023). The package also computes the maximum likelihood estimate for transition probabilities and estimates high-order MTD parameters through the expectation-maximization algorithm. Additionally, hdMTD also allows for simulating an MTD chain from its stationary invariant distribution using the perfect (exact) sampling algorithm, enabling Monte Carlo simulation of the model. We illustrate the package's capabilities through simulated data and a real-world application involving temperature records from Brazil.

stat.ME↗

arXiv2025

Self-switching random walks on Erdös-Rényi random graphs feel the phase transition

We study random walks on Erdös-Rényi random graphs in which, every time the random walk returns to the starting point, first an edge probability is independently sampled according to a priori measure $μ$, and then an Erdös-Rényi random graph is sampled according to that edge probability. When the edge probability $p$ does not depend on the size of the graph $n$ (dense case), we show that the proportion of time the random walk spends on different values of $p$ -- {\it occupation measure} -- converges to the a priori measure $μ$ as $n$ goes to infinity. More interestingly, when $p=λ/n$ (sparse case), we show that the occupation measure converges to a limiting measure with a density that is a function of the survival probability of a Poisson branching process. This limiting measure is supported on the supercritial values for the Erdös-Rényi random graphs, showing that self-witching random walks can detect the phase transition.

math.PR↗

arXiv2024

Structural results for the Tree Builder Random Walk

We study the Tree Builder Random Walk: a randomly growing tree, built by a walker as she is walking around the tree. Namely, at each time $n$, she adds a leaf to her current vertex with probability $p_n \asymp n^{-γ}$, $γ\in (2/3,1]$, then moves to a uniform random neighbor on the possibly modified tree. We show that the tree process at its growth times, after a random finite number of steps, can be coupled to be identical to the Barabási-Albert preferential attachment tree model. Thus, our TBRW-model is a local dynamics giving rise to the BA-model. The coupling also implies that many properties known for the BA-model, such as diameter and degree distribution, can be directly transferred to our TBRW-model, extending previous results.

math.PR↗

arXiv2024

A note on transience of generalized many-dimensional excited random walks

We consider a variation of the Generalized Excited Random Walk (GERW) in dimension $d\ge 2$ where the lower bound on the drift for excited jumps is time-dependent and decays to zero. We show that if the lower bound decays slower that $n^{-β}$ ($n$ is time), for $β$ depending on the transitions of the process, the GERW is transient in the direction of the drift.

math.PR↗

arXiv2023

Recurrence, transience and degree distribution for the Tree Builder Random Walk

We investigate a self-interacting random walk, whose dynamically evolving environment is a random tree built by the walker itself, as it walks around. At time $n=1,2,\dots$, right before stepping, the walker adds a random number (possibly zero) $Z_n$ of leaves to its current position. We assume that the $Z_n$'s are independent, but, importantly, we do \emph{not} assume that they are identically distributed. We obtain non-trivial conditions on their distributions under which the random walk is recurrent. This result is in contrast with some previous work in which, under the assumption that $Z_n\sim \mathsf{Ber}(p)$ (thus i.i.d.), the random walk was shown to be ballistic for every $p \in (0,1]$. We also obtain results on the transience of the walk, and the possibility that it ``gets stuck.'' From the perspective of the environment, we provide structural information about the sequence of random trees generated by the model when $Z_n\sim \mathsf{Ber}(p_n)$, with $p_n=Θ(n^{-γ})$ and $γ\in (2/3,1]$. We prove that the empirical degree distribution of this random tree sequence converges almost surely to a power-law distribution of exponent $3$, thus revealing a connection to the well known preferential attachment model.

math.PR↗

arXiv2023

Generalized Many-Dimensional Excited Random Walk in Bernoulli Environment

We study an extension of the generalized excited random walk (GERW) on $\mathbb{Z}^d$ introduced in [Ann. Probab. 40 (5), 2012, [7]] by Menshikov, Popov, Ramírez and Vachkovskaia. Our extension consists in studying a version of the GERW where excitation depends on a random environment. Given $p \in (0,1]$ (a parameter of the model) whenever the process visits a site for the first time, with probability $p$ it gains a drift in a given direction (could be any direction of the unit sphere). Otherwise, with probability $1-p$, it behaves as a $d$-martingale with zero-mean vector. Whenever the process visits an already-visited site, the process acts again as a $d$-martingale with zero-mean vector. We refer to the model as a GERW in Bernoulli environment, in short $p$-GERW. Under the same hypothesis of [7] (bounded jumps, uniform ellipticity), we show that the $p$-GERW is ballistic for all $p\in (0,1]$. Under the stronger assumptions that the increments of the regeneration times associated to the $p$-GERW are i.i.d. (condition which is satisfied, for example, for the excited random walk in a Bernoulli i.i.d. environment), we also obtain a Law of Large Numbers and a Central Limit Theorem.

math.PR↗

arXiv2022

A transient equivalence between Aldous-Broder and Wilson's algorithms and a two-stage framework for generating uniform spanning trees

The $Aldous\text{-}Broder$ and $Wilson$ are two well-known algorithms to generate uniform spanning trees (USTs) based on random walks. This work studies their relationship while they construct random trees with the goal of reducing the total time required to build the spanning tree. Using the notion of $branches$ $-$ paths generated by the two algorithms on particular stopping times, we show that the trees built by the two algorithms when running on a complete graph are statistically equivalent on these stopping times. This leads to a hybrid algorithm that can generate uniform spanning trees of complete graphs faster than either of the two algorithms. An efficient two-stage framework is also proposed to explore this hybrid approach beyond complete graphs, showing its feasibility in various examples, including transitive graphs where it requires 25% less time than $Wilson$ to generate a UST.

math.CO↗

arXiv2021

Tree Builder Random Walk: recurrence, transience and ballisticity

The Tree Builder Random Walk is a special random walk that evolves on trees whose size increases with time, randomly and depending upon the walker. After every s steps of the walker, a random number of vertices are added to the tree and attached to the current position of the walker. These processes share similarities with other important classes of markovian and non-markovian random walks presenting a large variety of behaviors according to parameters specifications. We show that for a large and most significant class of tree builder random walks, the process is either null recurrent or transient. If s is odd, the walker is ballistic and thus transient. If s is even, the walker's behavior can be explained from local properties of the growing tree and it can be either null recurrent or it gets trapped on some limited part of the growing tree.

math.PR↗

arXiv2019

The end time of SIS epidemics driven by random walks on edge-transitive graphs

Network epidemics is a ubiquitous model that can represent different phenomena and finds applications in various domains. Among its various characteristics, a fundamental question concerns the time when an epidemic stops propagating. We investigate this characteristic on a SIS epidemic induced by agents that move according to independent continuous time random walks on a finite graph: Agents can either be infected (I) or susceptible (S), and infection occurs when two agents with different epidemic states meet in a node. After a random recovery time, an infected agent returns to state S and can be infected again. The End of Epidemic (EoE) denotes the first time where all agents are in state S, since after this moment no further infections can occur and the epidemic stops. For the case of two agents on edge-transitive graphs, we characterize EoE as a function of the network structure by relating the Laplace transform of EoE to the Laplace transform of the meeting time of two random walks. Interestingly, this analysis shows a separation between the effect of network structure and epidemic dynamics. We then study the asymptotic behavior of EoE (asymptotically in the size of the graph) under different parameter scalings, identifying regimes where EoE converges in distribution to a proper random variable or to infinity. We also highlight the impact of different graph structures on EoE, characterizing it under complete graphs, complete bipartite graphs, and rings.

math.PR↗

arXiv2019

Building your path to escape from home

Random walks on dynamic graphs have received increasingly more attention from different academic communities over the last decade. Despite the relatively large literature, little is known about random walks that construct the graph where they walk while moving around. In this paper we study one of the simplest conceivable discrete time models of this kind, which works as follows: before every walker step, with probability $p$ a new leaf is added to the vertex currently occupied by the walker. The model grows trees and we call it the Bernoulli Growth Random Walk (BGRW). We show that the BGRW walker is transient and has a well-defined linear speed $c(p)>0$ for any $0<p\leq 1$. We also show that the tree as seen by the walker converges (in a suitable sense) to a random tree that is one-ended. Some natural open problems about this tree and variants of our model are collected at the end of the paper.

math.PR↗

arXiv2017

Transient and Slim versus Recurrent and Fat: Random Walks and the Trees they Grow

Network growth models that embody principles such as preferential attachment and local attachment rules have received much attention over the last decade. Among various approaches, random walks have been leveraged to capture such principles. In this paper we consider the No Restart Random Walk (NRRW) model where a walker builds its graph (tree) while moving around. In particular, the walker takes s steps (a parameter) on the current graph. A new node with degree one is added to the graph and connected to the node currently occupied by the walker. The walker then resumes, taking another s steps, and the process repeats. We analyze this process from the perspective of the walker and the network, showing a fundamental dichotomy between transience and recurrence for the walker as well as power law and exponential degree distribution for the network. More precisely, we prove the following results: i) for s=1 the random walk is transient and the degree of every node is bounded from above by a geometric distribution. ii) for s even the random walk is recurrent and the degree of non-leaf nodes is bounded from below by a power law distribution with exponent decreasing in s. We also provide a lower bound for the fraction of leaves in the graph, and for s=2 our bound implies that the fraction of leaves goes to one as the graph size goes to infinity. NRRW exhibits an interesting mutual dependency between graph building and random walking that is fundamentally influenced by the parity of s. Understanding this kind of coupled dynamics is an important step towards modeling more realistic network growth processes.

math.PR↗

arXiv2011

Potts model with invisible colours: Random-cluster representation and Pirogov-Sinai analysis

We study a variant of the ferromagnetic Potts model, recently introduced by Tamura, Tanaka and Kawashima, consisting of a ferromagnetic interaction among $q$ "visible" colours along with the presence of $r$ non-interacting "invisible" colours. We introduce a random-cluster representation for the model, for which we prove the existence of a first-order transition for any $q>0$, as long as $r$ is large enough. When $q>1$, the low-temperature regime displays a $q$-fold symmetry breaking. The proof involves a Pirogov-Sinai analysis applied to this random-cluster representation of the model.

math-ph↗

arXiv2011

First-order transition in Potts models with "invisible' states: Rigorous proofs

In some recent papers by Tamura, Tanaka and Kawashima [arXiv:1102.5475, arXiv:1012.4254], a class of Potts models with "invisible" states was introduced, for which the authors argued by numerical arguments and by a mean-field analysis that a first-order transition occurs. Here we show that the existence of this first-order transition can be proven rigorously, by relatively minor adaptations of existing proofs for ordinary Potts models. In our argument we present a random-cluster representation for the model, which might be of independent interest.

cond-mat.stat-mech↗

arXiv2011

A note on counting labeled and unlabeled trees

We provide a short combinatorial proof of Cayley's formula by means of a bijective map to an outcome space of an urn-drawing problem. Furthermore we introduce an algebraic structure on the set of labeled trees, which provides a more standard approach to Cayley's formula. Moreover, this algebraic structure sheds light on the problem of counting the unlabeled trees. In particular, it indicates how counting the number of unlabeled trees on $n$ vertices is connected to finding the number of partitions of $n-2$

math.CO↗

arXiv2010

Gibbs-non-Gibbs properties for evolving Ising models on trees

In this paper we study homogeneous Gibbs measures on a Cayley tree, subjected to an infinite-temperature Glauber evolution, and consider their (non-)Gibbsian properties. We show that the intermediate Gibbs state (which in zero field is the free-boundary-condition Gibbs state) behaves different from the plus and the minus state. E.g. at large times, all configurations are bad for the intermediate state, whereas the plus configuration never is bad for the plus state. Moreover, we show that for each state there are two transitions. For the intermediate state there is a transition from a Gibbsian regime to a non-Gibbsian regime where some, but not all configurations are bad, and a second one to a regime where all configurations are bad. For the plus and minus state, the two transitions are from a Gibbsian regime to a non-Gibbsian one and then back to a Gibbsian regime again.

math-ph↗

arXiv2010

Metastates in finite-type mean-field models: visibility, invisibility, and random restoration of symmetry

We consider a general class of disordered mean-field models where both the spin variables and disorder variables take finitely many values. To investigate the size-dependence in the phase-transition regime we construct the metastate describing the probabilities to find a large system close to a particular convex combination of the pure infinite-volume states. We show that, under a non-degeneracy assumption, only pure states are seen, with non-random probability weights for which we derive explicit expressions in terms of interactions and distributions of the disorder variables. We provide a geometric construction distinguishing invisible states (having zero weights) from visible ones. As a further consequence we show that, in the case where precisely two pure states are available, these must necessarily occur with the same weight, even if the model has no obvious symmetry relating the two.

math-ph↗