arXiv subjects

Matteo Sfragara

Publications and source records attributed to Matteo Sfragara.

12 recordsLinked to original sources

arXiv2026

Surviving from the tip of a cone in competing first-passage percolation

In two-type first passage percolation on $\mathbb{Z}^2$, two entities compete to capture the sites of the lattice. The entities spread between nearest neighbor sites at times specified by random passage times associated with the edges. We consider the case when both types have the same passage time distribution, with one type starting at the origin and the other from an infinite cone with tip at the origin and pointing in direction $\theta$. Itai Benjamini has suggested that the type starting at the origin can grow unboundedly if and only if the slope of the cone is strictly smaller than $\pi/2$, so that the cone does not fill a whole half-plane. The main result is that this is correct for any $\theta$ such that the asymptotic shape of the one-type process has a tangent line with direction $\theta$. The proofs are based on a description of infinite time-minimizing paths in terms of Busemann functions together with local modification arguments.

math.PR↗

arXiv2026

The Ising Model on a Two-Community Stochastic Block Model

We study the Ising model on a two-community stochastic block model, where $n$ spins are split into two equal groups with inter-community interaction parameter $\alpha_n\in[0,1]$. We provide a complete characterization of the phase diagram and show that, almost surely with respect to the graph realization, the model undergoes a uniqueness/non-uniqueness phase transition of the Gibbs measure. In particular, in the supercritical regime, the law of the magnetization vector of the two communities converges to a mixture of Dirac measures that, depending on whether $\alpha_n\gg 1/n$ or $\alpha_n\lesssim1/n$, is supported on two or four points, with possibly different weights. In the uniqueness region, we further analyze the fluctuations of the magnetization vector in the subcritical regime and we prove a quenched central limit theorem.

math.PR↗

arXiv2024

Mean field stable matchings

Consider the complete bipartite graph on $n+n$ vertices where the edges are equipped with i.i.d. exponential costs. A matching of the vertices is stable if it does not contain any pair of vertices where the connecting edge is cheaper than both matching costs. There exists a unique stable matching obtained by iteratively pairing vertices with small edge costs. We show that the total cost $C_{n,n}$ of this matching is of order $\log n$ with bounded variance, and that $C_{n,n}-\log n$ converges to a Gumbel distribution. We also show that the typical cost of an edge in the matching is of order $1/n$, with an explicit density on this scale, and analyze the rank of a typical edge. These results parallel those of Aldous for the minimal cost matching in the same setting. We then consider the sensitivity of the matching and the matching cost to perturbations of the underlying edge costs. The matching itself is shown to be robust in the sense that two matchings based on largely identical edge costs will have a substantial overlap. The matching cost however is shown to be noise sensitive, as a result of the fact that the most expensive edges will with high probability be replaced after resampling. Our proofs also apply to the complete (unipartite) graph and the results in this case are qualitatively similar.

math.PR↗

arXiv2023

Chaos, concentration and multiple valleys in first-passage percolation

A decade and a half ago Chatterjee established the first rigorous connection between anomalous fluctuations and a chaotic behaviour of the ground state in certain Gaussian disordered systems. The purpose of this paper is to show that Chatterjee's work gives evidence of a more general principle, by establishing an analogous connection between fluctuations and chaos in the context of first-passage percolation. The notion of `chaos' here refers to the sensitivity of the time-minimising path between two points when exposed to a slight perturbation. More precisely, we resample a small proportion of the edge weights, and find that a vanishing fraction of the edges on the time-minimising path still belongs to the time-minimising path obtained after resampling. We also identify the point at which the system transitions from being stable to being chaotic in terms of the variance of the system. Finally we show that the chaotic behaviour implies the existence of a large number of almost-optimal paths that are almost disjoint from the time-minimising path, a phenomenon known as 'multiple valleys'.

math.PR↗

arXiv2023

From stability to chaos in last-passage percolation

We study the transition from stability to chaos in a dynamic last passage percolation model on $\mathbb{Z}^d$ with random weights at the vertices. Given an initial weight configuration at time $0$, we perturb the model over time in such a way that the weight configuration at time $t$ is obtained by resampling each weight independently with probability $t$. On the cube $[0,n]^d$, we study geodesics, that is, weight-maximizing up-right paths from $(0,0, \dots, 0)$ to $(n,n, \dots, n)$, and their passage time $T$. Under mild conditions on the weight distribution, we prove a phase transition between stability and chaos at $t \asymp \frac{1}{n}\mathrm{Var}(T)$. Indeed, as $n$ grows large, for small values of $t$, the passage times at time $0$ and time $t$ are highly correlated, while for large values of $t$, the geodesics become almost disjoint.

math.PR↗

arXiv2022

The winner takes it all but one

We study competing first passage percolation on graphs generated by the configuration model with infinite-mean degrees. Initially, two uniformly chosen vertices are infected with type 1 and type 2 infection, respectively, and the infection then spreads via nearest neighbors in the graph. The time it takes for the type 1 (resp. 2) infection to traverse an edge $e$ is given by a random variable $X_1(e)$ (resp. $X_2(e)$) and, if the vertex at the other end of the edge is still uninfected, it then becomes type 1 (resp. 2) infected and immune to the other type. Assuming that the degrees follow a power-law distribution with exponent $\tau \in (1,2)$, we show that, with high probability as the number of vertices tends to infinity, one of the infection types occupies all vertices except for the starting point of the other type. Moreover, both infections have a positive probability of winning regardless of the passage times distribution. The result is also shown to hold for the erased configuration model, where self-loops are erased and multiple edges are merged, and when the degrees are conditioned to be smaller than $n^\alpha$ for some $\alpha > 0$.

math.PR↗

arXiv2021

Adding edge dynamics to bipartite random-access networks

We consider random-access networks with nodes representing transmitter-receiver pairs whose signals interfere with each other depending on their vicinity. Data packets arrive at the nodes over time and form queues. The nodes can be either active or inactive: a node deactivates at unit rate, while it activates at a rate that depends on its queue length, provided none of its neighbors is active. In order to model the effects of user mobility in wireless networks, we analyze dynamic interference graphs where the edges are allowed to appear and disappear over time. We focus on bipartite graphs and study the transition time between the two states where one part of the network is active and the other part is inactive, in the limit as the queue lengths become large. Depending on the speed of the dynamics, we are able to obtain a rough classification of the effects of the dynamics on the transition time.

math.PR↗

arXiv2020

Large deviation principle for the maximal eigenvalue of inhomogeneous Erd\H{o}s-R\'enyi random graphs

We consider an inhomogeneous Erd\H{o}s-R\'enyi random graph $G_N$ with vertex set $[N] = \{1,\dots,N\}$ for which the pair of vertices $i,j \in [N]$, $i\neq j$, is connected by an edge with probability $r(\tfrac{i}{N},\tfrac{j}{N})$, independently of other pairs of vertices. Here, $r\colon\,[0,1]^2 \to (0,1)$ is a symmetric function that plays the role of a reference graphon. Let $\lambda_N$ be the maximal eigenvalue of the adjacency matrix of $G_N$. It is known that $\lambda_N/N$ satisfies a large deviation principle as $N \to \infty$. The associated rate function $\psi_r$ is given by a variational formula that involves the rate function $I_r$ of a large deviation principle on graphon space. We analyse this variational formula in order to identify the properties of $\psi_r$, specially when the reference graphon is of rank 1.

math.PR↗

arXiv2020

Wireless random-access networks with bipartite interference graphs

We consider random-access networks where nodes represent servers with a queue and can be either active or inactive. A node deactivates at unit rate, while it activates at a rate that depends on its queue length, provided none of its neighbors is active. We consider arbitrary bipartite graphs in the limit as the initial queue lengths become large and identify the transition time between the two states where one half of the network is active and the other half is inactive. The transition path is decomposed into a succession of transitions on complete bipartite subgraphs. We formulate a randomized greedy algorithm that takes the graph as input and gives as output the set of transition paths the network is most likely to follow. Along each path we determine the mean transition time and its law on the scale of its mean. Depending on the activation rates, we identify three regimes of behavior.

math.PR↗

arXiv2018

Spectra of Adjacency and Laplacian Matrices of Inhomogeneous Erd\H{o}s-R\'enyi Random Graphs

Inhomogeneous Erd\H{o}s-R\'enyi random graphs $\mathbb G_N$ on $N$ vertices in the non-dense regime are considered in this paper. The edge between the pair of vertices $\{i,j\}$ is retained with probability $\varepsilon_N\,f(\frac{i}{N},\frac{j}{N})$, $1 \leq i \neq j \leq N$, independently of other edges, where $f\colon\,[0,1] \times [0,1] \to [0,\infty)$ is a continuous function such that $f(x,y)=f(y,x)$ for all $x,y \in [0,1]$. We study the empirical distribution of both the adjacency matrix $A_N$ and the Laplacian matrix $\Delta_N$ associated with $\mathbb G_N$ in the limit as $N \to \infty$ when $\lim_{N\to\infty} \varepsilon_N = 0$ and $\lim_{N\to\infty} N\varepsilon_N = \infty$. In particular, it is shown that the empirical spectral distributions of $A_N$ and $\Delta_N$, after appropriate scaling and centering, converge to deterministic limits weakly in probability. For the special case where $f(x,y) = r(x)r(y)$ with $r\colon\,[0,1] \to [0,\infty)$ a continuous function, we give an explicit characterization of the limiting distributions. Furthermore, applications of the results to constrained random graphs, Chung-Lu random graphs and social networks are shown.

math.PR↗

arXiv2018

Transition time asymptotics of queue-based activation protocols in random-access networks

We consider networks where each node represents a server with a queue. An active node deactivates at unit rate. An inactive node activates at a rate that depends on its queue length, provided none of its neighbors is active. For complete bipartite networks, in the limit as the queues become large, we compute the average transition time between the two states where one half of the network is active and the other half is inactive. We show that the law of the transition time divided by its mean exhibits a trichotomy, depending on the activation rate functions.

math.PR↗

arXiv2017

The switch Markov chain for sampling irregular graphs and digraphs

The problem of efficiently sampling from a set of (undirected, or directed) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the sampling. The switch chain is known to be rapidly mixing for regular degree sequences, both in the undirected and directed setting. We prove that the switch chain for undirected graphs is rapidly mixing for any degree sequence with minimum degree at least 1 and with maximum degree $d_{\max}$ which satisfies $3\leq d_{\max}\leq \frac{1}{3}\, \sqrt{M}$, where $M$ is the sum of the degrees. The mixing time bound obtained is only a factor $n$ larger than that established in the regular case, where $n$ is the number of vertices. Our result covers a wide range of degree sequences, including power-law graphs with parameter $\gamma > 5/2$ and sufficiently many edges. For directed degree sequences such that the switch chain is irreducible, we prove that the switch chain is rapidly mixing when all in-degrees and out-degrees are positive and bounded above by $\frac{1}{4}\, \sqrt{m}$, where $m$ is the number of arcs, and not all in-degrees and out-degrees equal 1. The mixing time bound obtained in the directed case is an order of $m^2$ larger than that established in the regular case.

cs.DM↗