arXiv subjects

Umberto De Ambroggio

Publications and source records attributed to Umberto De Ambroggio.

16 recordsLinked to original sources

arXiv2026

On the phase transition for the number of collisions on comb graphs

We consider collisions of simple random walks on comb graphs $\mathrm{Comb}(\mathbb{Z},H)$, which are obtained by attaching vertical segments of the form $[0,H_x] \cap \mathbb{Z}$ to any point $x$ of the integer axis. For $\mathrm{Comb}(\mathbb{Z},H)$ with profile $H_x(x) = |x| \log^γ(|x| \vee 1)$, we show that two independent simple random walks starting from the same site collide infinitely often almost surely if $γ\leq 2$. If the tooth profile is taken as a typical realization of i.i.d. heavy-tailed random variables with $\textbf{P}(H_x > z) \sim Cz^{-γ}$ (with some $C > 0$) as $z$ tends to infinity, we show that infinitely many collisions occur almost surely for two independent random walks if $γ> 1/3$, whereas finitely many collisions occur almost surely if $γ\in (0,1/3)$, and for any $γ\in (0,1]$, three independent random walks only collide finitely many times, almost surely.

math.PR↗

arXiv2026

Very large cliques in a scale-free random graph

In this short article we consider a preferential attachment random graph model with edge steps, studied by Alves, Ribeiro and Sanchis. Starting with an initial graph $\mathbb{G}_1$ formed by a vertex with a self-loop attached to it, the model evolves as follows. At every subsequent (discrete) time step, either with probability $p$ we add a vertex to the graph and connect it to exactly one of the older vertices selected with probability proportional to its degree, or with probability $1-p$ we add one edge between two existing vertices, both selected (independently) with probability proportional to their degrees. Let $ω(\mathbb{G})$ be the clique number of a graph $\mathbb{G}$, i.e.\ the number of vertices in a largest complete subgraph of $\mathbb{G}_{}$. Alves, Ribeiro and Sanchis showed that, for any given $\varepsilon>0$, we have $ω(\mathbb{G}_{2t})\geq t^{\frac{1-p}{2-p}(1-\varepsilon)}$ with high probability (i.e.\ with probability tending to $1$ as $t\rightarrow \infty$). Here we strengthen this bound by showing that, for any function $f:\mathbb{N}\mapsto \mathbb{N}$ that satisfies $f(t)\rightarrow \infty$ as $t\rightarrow \infty$, with high probability \[ω(\mathbb{G}_{2t}) = Ω\left(t^{\frac{1-p}{2-p}}\Big(\log^{\frac{1}{2-p}}(t)f(t)\Big)^{-1}\right).\]

math.PR↗

arXiv2025

Repeat times and a two-weight UST model

We study a model of random weighted uniform spanning trees on the complete graph with $n$ vertices, where each edge is assigned a weight of $n^{1+γ}$ with probability $1/n$ and $1$ otherwise. Whenever $γ$ is large enough, we prove that the diameter of the resulting tree is typically of order $n^{1/3} \log n$, up to a $\log \log n$ correction. Our approach uses estimates on repeat times for selecting components in a critical Erdős-Rényi graph, as well as concentration bounds on the sums of diameters of these components.

math.PR↗

arXiv2025

Collisions of random walks on comb graphs with a planar base

In this article we study collisions of two independent random walks on comb graphs $\mathrm{Comb}(\tilde{G},f)$ for a large class of recurrent planar graphs $\tilde{G}$ and profile functions $f$, the latter governing the length of vertical segments (called "teeth") attached to vertices of the base graph $\tilde{G}$. We prove that the number of collisions of two random walks starting from the same site undergoes a phase transition depending on the growth of $f$. As a benchmark example, we show that for $\mathrm{Comb}(\mathbb{Z}^2,f_γ)$ with $f_γ(z) = \log^γ(\|z\|_\infty\vee 1)$ and $\|\cdot \|_\infty$ denoting the supremum norm, two independent random walks started at the origin collide finitely often almost surely if $γ> 1$, answering a question of Barlow, Peres, and Sousi, see arXiv:1003.3255, who established that infinitely many collisions occur almost surely if $γ\leq 1$. We furthermore establish phase transitions in the cases where the base graph $\tilde{G}$ is pre-fractal, or a typical realization of a supercritical cluster of planar Bernoulli bond percolation.

math.PR↗

arXiv2025

Quenched scaling limit for biased random walks on random, heavy tailed conductances: low dimensions

We consider a random walk amongst positive random conductances on $\mathbb{Z}^d, d \ge 2$, with directional bias. When the conductances have a stable distribution with parameter $γ\in (0, 1)$, the walk is sub-ballistic. In this regime Fribergh and Kious (Ann. Prob. 2018) derived an annealed scaling limit for the appropriately rescaled walk towards the Fractional Kinetics process. We prove the quenched version of this result for all $d \ge 2$.

math.PR↗

arXiv2025

Small maximal clusters are very unlikely in critical random graphs

We describe a probabilistic methodology, based on random walk estimates, to obtain exponential upper bounds for the probability of observing unusually small maximal components in two classical (near-)critical random graph models. More specifically, we analyse the near-critical Erdős-Rényi model $\mathbb{G}(n,p)$ and the random graph $\mathbb{G}(n,d,p)$ obtained by performing near-critical $p$-bond percolation on a simple random $d$-regular graph and show that, for each one of these models, the probability that the size of a largest component is smaller than $n^{2/3}/A$ is at most of order $\exp(-A^{3/2})$. The exponent $3/2$ is known to be optimal for the near-critical $\mathbb{G}(n,p)$ random graph, whereas for the near-critical $\mathbb{G}(n,d,p)$ model the best known upper bound for the above probability was of order $A^{-3/5}$. As a secondary result we show, by means of an optimized version of the martingale method of Nachmias and Peres, that the above probability of observing an unusually small maximal component is at most of order $\exp(-A^{3/5})$ in other two critical models, namely a random intersection graph and the quantum random graph; this stretched-exponential bounds also improve upon the known (polynomial) bounds available for these other two critical models.

math.PR↗

arXiv2024

Dispersion on the Complete Graph

We consider a synchronous process of particles moving on the vertices of a graph $G$, introduced by Cooper, McDowell, Radzik, Rivera and Shiraga (2018). Initially, $M$ particles are placed on a vertex of $G$. At the beginning of each time step, for every vertex inhabited by at least two particles, each of these particles moves independently to a neighbour chosen uniformly at random. The process ends at the first step when no vertex is inhabited by more than one particle. Cooper et al. showed that when the underlying graph is the complete graph on~$n$ vertices, then there is a phase transition when the number of particles $M = n/2$. They showed that if $M<(1-\varepsilon)n/2$ for some fixed $\varepsilon>0$, then the process finishes in a logarithmic number of steps, while if $M>(1+\varepsilon)n/2$, an exponential number of steps are required with high probability. Here we provide a thorough asymptotic analysis of the dispersion time around criticality, where $\varepsilon = o(1)$, and describe the transition from logarithmic to exponential time. As a consequence of our results we establish, for example, that the dispersion time is in probability and in expectation in $Θ(n^{1/2})$ when $|\varepsilon| = O(n^{-1/2})$, and provide qualitative bounds for its tail behavior.

math.PR↗

arXiv2024

Balanced Bidirectional Breadth-First Search on Scale-Free Networks

To find a shortest path between two nodes $s_0$ and $s_1$ in a given graph, a classical approach is to start a Breadth-First Search (BFS) from $s_0$ and run it until the search discovers $s_1$. Alternatively, one can start two Breadth-First Searches, one from $s_0$ and one from $s_1$, and alternate their layer expansions until they meet. This bidirectional BFS can be balanced by always expanding a layer on the side that has discovered fewer vertices so far. This usually results in significant speedups in real-world networks, and it has been shown that this indeed yields sublinear running time on scale-free graph models such as Chung-Lu graphs and hyperbolic random graphs. We improve this layer-balanced bidirectional BFS approach by using a finer balancing technique. Instead of comparing the size of the two BFS trees after each layer expansion, we perform this comparison after each vertex expansion. This gives rise to two algorithms that run faster than the layer-balanced bidirectional BFS on scale-free networks with power-law exponent $τ\in (2,3)$. The first one is an approximate shortest-path algorithm that outputs a path of length at most 1 longer than the shortest path in time $n^{(τ-2)/(τ-1)+o(1)}$. The second one is an exact shortest-path algorithm running in time $n^{1/2+o(1)}$. These runtime bounds hold with high probability when $s_0$ and $s_1$ are chosen uniformly at random among the $n$ vertices of the graph. We also develop an edge-balanced bidirectional BFS algorithm that works under adversarial conditions. This approximate shortest-path algorithm runs in time $n^{1/2+o(1)}$ with high probability when the adversary is allowed to choose $s_0$ and $s_1$ based on their (expected) degree. We complement our theoretical results with experiments on Chung-Lu graphs, Geometric Inhomogeneous Random Graphs, and real-world networks.

cs.SI↗

arXiv2024

Triple collisions on a comb graph

In this article, we consider the number of collisions of three independent simple random walks on a subgraph of the two-dimensional square lattice obtained by removing all horizontal edges with vertical coordinate not equal to 0 and then, for $n\in \mathbb{Z}$, restricting the vertical segment of the graph located at horizontal coordinate $n$ to the interval $\{0,1,\dots,\log^α(|n|\vee 1)\}$. Specifically, we show the following phase transition: when $α\leq 1$, the three random walks collide infinitely many times almost-surely, whereas when $α>1$, they collide only finitely many times almost-surely. This is a variation of a result of Barlow, Peres and Sousi, who showed a similar phase transition for two random walks when the vertical segments are truncated at height $|n|^α$.

math.PR↗

arXiv2024

Limit Laws for Critical Dispersion on Complete Graphs

We consider a synchronous process of particles moving on the vertices of a graph $G$, introduced by Cooper, McDowell, Radzik, Rivera and Shiraga (2018). Initially, $M$ particles are placed on a vertex of $G$. In subsequent time steps, all particles that are located on a vertex inhabited by at least two particles jump independently to a neighbour chosen uniformly at random. The process ends at the first step when no vertex is inhabited by more than one particle; we call this (random) time step the dispersion time. In this work we study the case where $G$ is the complete graph on $n$ vertices and the number of particles is $M=n/2+αn^{1/2} + o(n^{1/2})$, $α\in \mathbb{R}$. This choice of $M$ corresponds to the critical window of the process, with respect to the dispersion time. We show that the dispersion time, if rescaled by $n^{-1/2}$, converges in $p$-th mean, as $n\rightarrow \infty$ and for any $p \in \mathbb{R}$, to a continuous and almost surely positive random variable $T_α$. We find that $T_α$ is the absorption time of a standard logistic branching process, thoroughly investigated by Lambert (2005), and we determine its expectation. In particular, in the middle of the critical window we show that $\mathbb{E}[T_0] = π^{3/2}/\sqrt{7}$, and furthermore we formulate explicit asymptotics when $|α|$ gets large that quantify the transition into and out of the critical window. We also study the (random) total number of jumps that are performed by the particles until the dispersion time is reached. In particular, we prove that it centers around $\frac{2}{7}n\ln n$ and that it has variations linear in $n$, whose distribution we can describe explicitly.

math.PR↗

arXiv2023

Upper bounds for the largest components in critical inhomogeneous random graphs

We consider the Norros-Reittu random graph $NR_n(\textbf{w})$, where edges are present independently but edge probabilities are moderated by vertex weights, and use probabilistic arguments based on martingales to analyse the component sizes in this model when considered at criticality. In particular, we obtain stronger upper bounds (with respect to those available in the literature) for the probability of observing unusually large maximal clusters, and simplify the arguments needed to derive polynomial upper bounds for the probability of observing unusually small largest components.

math.PR↗

arXiv2023

A simple path to component sizes in critical random graphs

We describe a robust methodology, based on the martingale argument of Nachmias and Peres and random walk estimates, to obtain simple upper and lower bounds on the size of a maximal component in several random graphs \textit{at criticality}. Even though the main result is not new, we believe the the material presented here is interesting because it unifies several proofs found in the literature into a common framework. More specifically, we give easy-to-check conditions that, when satisfied, allow an immediate derivation of the above mentioned bounds.

math.PR↗

arXiv2022

An elementary approach to component sizes in critical random graphs

In this article we introduce a simple tool to derive polynomial upper bounds for the probability of observing unusually large maximal components in some models of random graphs when considered at criticality. Specifically, we apply our method to a model of random intersection graph, a random graph obtained through $p$-bond percolation on a general $d$-regular graph, and a model of inhomogeneous random graph.

math.PR↗

arXiv2021

The probability of unusually large components for critical percolation on random $d$-regular graphs

Let $d\ge 3$ be a fixed integer, $p\in (0,1)$, and let $n\geq 1$ be a positive integer such that $dn$ is even. Let $\mathbb{G}(n, d, p)$ be a (random) graph on $n$ vertices obtained by drawing uniformly at random a $d$-regular (simple) graph on $[n]$ and then performing independent $p$-bond percolation on it, i.e. we independently retain each edge with probability $p$ and delete it with probability $1-p$. Let $|\mathcal{C}_{\text{max}}|$ be the size of the largest component in $\mathbb{G}(n, d, p)$. We show that, when $p$ is of the form $p=(d-1)^{-1}(1+λn^{-1/3})$ for $λ\in \mathbb{R}$, and $A$ is large, \begin{align*} \mathbb{P}(|\mathcal{C}_{\text{max}}|>An^{2/3})\asymp A^{-3/2}e^{-\frac{A^3(d-1)(d-2)}{8d^2}+\frac{λA^2(d-2)^2}{2d(d-1)}-\frac{λ^2 A(d-1)}{2(d-2)}}. \end{align*} This improves on a result of Nachmias and Peres. We also give an analogous asymptotic for the probability that a particular vertex is in a component of size larger than $An^{2/3}$.

math.PR↗

arXiv2021

Unusually large components in near-critical Erdős-Rényi graphs via ballot theorems

We consider the near-critical Erdős-Rényi random graph $G(n,p)$ and provide a new probabilistic proof of the fact that, when $p$ is of the form $p=p(n)=1/n+λ/n^{4/3}$ and $A$ is large, \[\mathbb{P}(|\mathcal{C}_{\max}|>An^{2/3})\asymp A^{-3/2}e^{-\frac{A^3}{8}+\frac{λA^2}{2}-\frac{λ^2A}{2}}\] where $\mathcal{C}_{\max}$ is the largest connected component of the graph. Our result allows $A$ and $λ$ to depend on $n$. While this result is already known, our proof relies only on conceptual and adaptable tools such as ballot theorems, whereas the existing proof relies on a combinatorial formula specific to Erdős-Rényi graphs, together with analytic estimates.

math.PR↗

arXiv2020

On dynamic random graphs with degree homogenization via anti-preferential attachment probabilities

We analyze a dynamic random undirected graph in which newly added vertices are connected to those already present in the graph either using, with probability $p$, an anti-preferential attachment mechanism or, with probability $1-p$, a preferential attachment mechanism. We derive the asymptotic degree distribution in the general case and study the asymptotic behaviour of the expected degree process in the general and that of the degree process in the pure anti-preferential attachment case. Degree homogenization mainly affects convergence rates for the former case and also the limiting degree distribution in the latter.

math.PR↗