arXiv subjects

Daniel Valesin

Publications and source records attributed to Daniel Valesin.

At least 19 recordsLinked to original sources

arXiv2026

Ergodicity of the voter model with dynamic anti-voter bonds

The voter model with anti-voter bonds is a variant of the classical voter model in which the edges of the underlying graph are assigned signs. At each update, a voter chooses a neighbour according to a transition kernel; interactions across a positive edge follow the usual voter dynamics, so that a site adopts the current opinion of its chosen neighbour, whereas interactions across a negative edge lead to the adoption of the opposite opinion. In this work, we introduce a new variant in which the edge signs evolve dynamically according to dynamical percolation with density parameter $p \in (0,1)$ and speed $\mathsf{v} \in (0,\infty)$, where the two states of the process represent positive and negative edges. This defines a joint spin-bond Markov process. Following Liggett's notion of ergodicity, we prove that this process is ergodic on any simple graph with countably many vertices, with an arbitrary transition kernel of adoption rates and for all choices of the parameters of the edge dynamics.

math.PR↗

arXiv2026

The contact process on a bipartite spatial network

We study the contact process on a random bipartite connection hypergraph generated from two Poisson point processes, with mark-dependent connection thresholds. For asymmetric infection rates and asymmetric power law tail decays of the two degree distributions, we determine the dominant survival strategies in all parameter regimes and provide asymptotics for the epidemic probability up to logarithmic factors.

math.PR↗

arXiv2026

Interface for variants of the contact process

We study two one-dimensional variants of the contact process: the contact-and-barrier process, where the population evolves in a region delimited by a randomly moving barrier, and the multitype contact process, in which two species compete for space. The contact-and-barrier process is started with the barrier at the origin and all sites to its right occupied, while the multitype contact process is started from the Heaviside configuration with species 1 to the left of the origin and species 2 to the right. We prove that both models exhibit tight interfaces and that, after centring by an appropriate deterministic speed, the interface position satisfies a central limit theorem. Our analysis relies on a renewal-time method based on a novel construction called patchwork construction, in which the processes are built by concatenating space-time evolutions over successive time intervals of random length, providing a more convenient framework for defining the renewal times that drive the proofs.

math.PR↗

arXiv2024

Percolation on the stationary distributions of the voter model with stirring

The voter model with stirring is a variant of the classical voter model on $\mathbb{Z}^d$ with two possible opinions (0 and 1) that, in addition to copying neighbouring opinions at rate 1, allows voters to interchange their opinions at rate~$\mathsf{v}$ where~$\mathsf v \ge 0$ is the stirring parameter. This model was considered in \cite{Astoquillca24}, where it was proved that for~$d \ge 3$ and for any~$\mathsf{v}$ the set of extremal stationary measures is given by a family~$\{ \mu_{\alpha,\mathsf{v}}: \alpha \in [0,1] \}$, where~$\alpha$ is the density of voters with opinion~1. Sampling a configuration~$\xi$ from~$\mu_{\alpha, \mathsf v}$, we study~$\xi$ as a site percolation model on~$\mathbb{Z}^d$, where the set of occupied sites is the set of voters with opinion 1 in~$\xi$. Letting~$\alpha_c(\mathsf v)$ be the supremum of all the values of~$\alpha$ for which percolation does not occur~$\mu_{\alpha, \mathsf v}$-a.s., we prove that $\alpha_c(\mathsf{v})$ converges to~$p_c$, the critical density for classical Bernoulli site percolation, as~$\mathsf{v}$ tends to infinity. As a consequence, for $\mathsf v$ large enough, the model exhibits a non-trivial phase transition in~$\alpha$.

math.PR↗

arXiv2024

Speed of Convergence and Moderate Deviations of FPP on Random Geometric Graphs

This study delves into first-passage percolation on random geometric graphs in the supercritical regime, where the graphs exhibit a unique infinite connected component. We investigate properties such as geodesic paths, moderate deviations, and fluctuations, aiming to establish a quantitative shape theorem. Furthermore, we examine fluctuations in geodesic paths and characterize the properties of spanning trees and their semi-infinite paths.

math.PR↗

arXiv2024

Catalan percolation

In Catalan percolation, all nearest-neighbor edges $\{i,i+1\}$ along $\mathbb Z$ are initially occupied, and all other edges are open independently with probability $p$. Open edges $\{i,j\}$ are occupied if some pair of edges $\{i,k\}$ and $\{k,j\}$, with $i<k<j$, become occupied. This model was introduced by Gravner and the third author, in the context of polluted graph bootstrap percolation. We prove that the critical $p_{\mathrm c}$ is strictly between that of oriented site percolation on $\mathbb Z^2$ and the Catalan growth rate $1/4$. Our main result shows that an enhanced oriented percolation model, with non-decaying infinite-range dependency, has a strictly smaller critical parameter than the classical model. This is reminiscent of the work of Duminil-Copin, Hil\'ario, Kozma and Sidoravicius on brochette percolation. Our proof differs, however, in that we do not use Aizenman--Grimmett enhancements or differential inequalities. Two key ingredients are the work of Hil\'ario, S\'a, Sanchis and Teixeira on stretched lattices, and the Russo--Seymour--Welsh result for oriented percolation by Duminil-Copin, Tassion and Teixeira.

math.PR↗

arXiv2023

Degree-penalized contact processes

We study degree-penalized contact processes on Galton-Watson trees (GW) and the configuration model. The model we consider is a modification of the usual contact process on a graph. In particular, each vertex can be either infected or healthy. When infected, each vertex heals at rate one. Also, when infected, a vertex $v$ with degree $d_v$ infects its neighboring vertex $u$ with degree $d_u$ with rate $\lambda/f(d_u, d_v)$ for some positive function $f$. In the case $f(d_u, d_v)=\max(d_u, d_v)^\mu$ for some $\mu>0$, the infection is slowed down to and from high degree vertices. This is in line with arguments used in social network science: people with many contacts do not have the time to infect their neighbors at the same rate as people with fewer contacts. We show that new phase transitions occur in terms of the parameter $\mu$ (at $1/2$) and the degree distribution $D$ of the GW tree. - When $\mu\ge 1$, the process goes extinct for all distributions $D$ for all sufficiently small $\lambda>0$; - When $\mu\in(1/2, 1)$, and the tail of $D$ weakly follows a power law with tail-exponent less than $1-\mu$, the process survives globally but not locally for all $\lambda$ small enough; - When $\mu\in(1/2, 1)$, and $\mathbb{E}[D^{1-\mu}]<\infty$, the process goes extinct almost surely, for all $\lambda$ small enough; - When $\mu<1/2$, and $D$ is heavier then stretched exponential with stretch-exponent $1-2\mu$, the process survives (locally) with positive probability for all $\lambda>0$. We also study the product case $f(x,y)=(xy)^\mu$. In that case, the situation for $\mu < 1/2$ is the same as the one described above, but $\mu\ge 1/2$ always leads to a subcritical contact process for small enough $\lambda>0$ on all graphs. Furthermore, for finite random graphs with prescribed degree sequences, we establish the corresponding phase transitions in terms of the length of survival.

math.PR↗

arXiv2023

The contact process on dynamic regular graphs: monotonicity and subcritical phase

We study the contact process on a dynamic random~$d$-regular graph with an edge-switching mechanism, as well as an interacting particle system that arises from the local description of this process, called the herds process. Both these processes were introduced in~\cite{da2021contact}; there it was shown that the herds process has a phase transition with respect to the infectivity parameter~$λ$, depending on the parameter~$\mathsf{v}$ that governs the edge dynamics. Improving on a result of~\cite{da2021contact}, we prove that the critical value of~$λ$ is strictly decreasing with~$\mathsf{v}$. We also prove that in the subcritical regime, the extinction time of the herds process started from a single individual has an exponential tail. Finally, we apply these results to study the subcritical regime of the contact process on the dynamic $d$-regular graph. We show that, starting from all vertices infected, the infection goes extinct in a time that is logarithmic in the number of vertices of the graph, with high probability.

math.PR↗

arXiv2023

Scaling limit of an adaptive contact process

We introduce and study an interacting particle system evolving on the $d$-dimensional torus $(\mathbb Z/N\mathbb Z)^d$. Each vertex of the torus can be either empty or occupied by an individual of type $λ\in (0,\infty)$. An individual of type $λ$ dies with rate one and gives birth at each neighboring empty position with rate $λ$; moreover, when the birth takes place, the newborn individual is likely to have the same type as the parent, but has a small probability of being a mutant. A mutant child of an individual of type $λ$ has type chosen according to a probability kernel. We consider the asymptotic behavior of this process when $N\to \infty$ and the parameter $δ_N$ tends to zero fast enough that mutations are sufficiently separated in time, so that the amount of time spent on configurations with more than one type becomes negligible. We show that, after a suitable time scaling and deletion of the periods of time spent on configurations with more than one type, the process converges to a Markov jump process on $(0,\infty)$, whose rates we characterize.

math.PR↗

arXiv2023

Limiting shape for First-Passage Percolation models on Random Geometric Graphs

Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variables on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape and we show that the shape is an Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. For the Richardson model we further show that it converges weakly to a nonstandard branching process in the joint limit of large intensities and slow passage times.

math.PR↗

arXiv2022

Face and cycle percolation

We consider face and cycle percolation as models for continuum percolation based on random simplicial complexes in Euclidean space. Face percolation is defined through infinite sequences of $d$-simplices sharing a $(d-1)$-dimensional face. In contrast, cycle percolation demands the existence of infinite $d$-cycles, thereby generalizing the lattice notion of plaquette percolation. We discuss the sharp phase transition for face percolation and derive comparison results between the critical intensities for face and cycle percolation. Finally, we consider an alternate version of simplex percolation, by declaring simplices to be neighbors whenever they are sufficiently close to each other, and prove a strict inequality involving the critical intensity of this alternate version and that of face percolation.

math.PR↗

arXiv2021

The contact process over a dynamical d-regular graph

We consider the contact process on a dynamic graph defined as a random $d$-regular graph with a stationary edge-switching dynamics. In this graph dynamics, independently of the contact process state, each pair $\{e_1,e_2\}$ of edges of the graph is replaced by new edges $\{e_1',e_2'\}$ in a crossing fashion: each of $e_1',e_2'$ contains one vertex of $e_1$ and one vertex of $e_2$. As the number of vertices of the graph is taken to infinity, we scale the rate of switching in a way that any fixed edge is involved in a switching with a rate that approaches a limiting value $\mathsf{v}$, so that locally the switching is seen in the same time scale as that of the contact process. We prove that if the infection rate of the contact process is above a threshold value $\barλ$ (depending on $d$ and $\mathsf{v}$), then the infection survives for a time that grows exponentially with the size of the graph. By proving that $\barλ$ is strictly smaller than the lower critical infection rate of the contact process on the infinite $d$-regular tree, we show that there are values of $λ$ for which the infection dies out in logarithmic time in the static graph but survives exponentially long in the dynamic graph.

math.PR↗

arXiv2021

Reaction-diffusion models for a class of infinite-dimensional non-linear stochastic differential equations

We establish the existence of solutions to a class of non-linear stochastic differential equation of reaction-diffusion type in an infinite-dimensional space, with diffusion corresponding to a given transition kernel. The solution obtained is the scaling limit of a sequence of interacting particle systems, and satisfies the martingale problem corresponding to the target differential equation.

math.PR↗

arXiv2021

Results on the contact process with dynamic edges or under renewals

We analyze variants of the contact process that are built by modifying the percolative structure given by the graphical construction and develop a robust renormalization argument for proving extinction in such models. With this method, we obtain results on the phase diagram of two models: the Contact Process on Dynamic Edges introduced by Linker and Remenik and a generalization of the Renewal Contact Process introduced by Fontes, Marchetti, Mountford and Vares.

math.PR↗

arXiv2021

On the threshold of spread-out contact process percolation

We study the stationary distribution of the (spread-out) $d$-dimensional contact process from the point of view of site percolation. In this process, vertices of $\mathbb{Z}^d$ can be healthy (state 0) or infected (state 1). With rate one infected sites recover, and with rate $λ$ they transmit the infection to some other vertex chosen uniformly within a ball of radius $R$. The classical phase transition result for this process states that there is a critical value $λ_c(R)$ such that the process has a non-trivial stationary distribution if and only if $λ> λ_c(R)$. In configurations sampled from this stationary distribution, we study nearest-neighbor site percolation of the set of infected sites; the associated percolation threshold is denoted $λ_p(R)$. We prove that $λ_p(R)$ converges to $1/(1-p_c)$ as $R$ tends to infinity, where $p_c$ is the threshold for Bernoulli site percolation on $\mathbb{Z}^d$. As a consequence, we prove that $λ_p(R) > λ_c(R)$ for large enough $R$, answering an open question of Liggett and Steif in the spread-out case.

math.PR↗

arXiv2021

Approximation on slabs and uniqueness for inhomogeneous percolation with a plane of defects

Let $ \mathbb{L}^{d} = ( \mathbb{Z}^{d},\mathbb{E}^{d} ) $ be the $ d $-dimensional hypercubic lattice. We consider a model of inhomogeneous Bernoulli percolation on $ \mathbb{L}^{d} $ in which every edge inside the $ s $-dimensional hyperplane $ \mathbb{Z}^{s} \times \{ 0 \}^{d-s} $, $ 2 \leq s < d $, is open with probability $ q $ and every other edge is open with probability $ p $. We prove the uniqueness of the infinite cluster in the supercritical regime whenever $ p \neq p_{c}(d) $, where $ p_{c}(d) $ denotes the threshold for homogeneous percolation, and that the critical point $ (p,q_{c}(p)) $ can be approximated on the phase space by the critical points of slabs, for any $ p < p_{c}(d) $.

math.PR↗

arXiv2021

Multi-range percolation on oriented trees: critical curve and limit behavior

We consider an inhomogeneous oriented percolation model introduced by de Lima, Rolla and Valesin. In this model, the underlying graph is an oriented rooted tree in which each vertex points to each of its $d$ children with `short' edges, and in addition, each vertex points to each of its $d^k$ descendant at a fixed distance $k$ with `long' edges. A bond percolation process is then considered on this graph, with the prescription that independently, short edges are open with probability $p$ and long edges are open with probability $q$. We study the behavior of the critical curve $q_c(p)$: we find the first two terms in the expansion of $q_c(p)$ as $k \to \infty$, and prove that the critical curve lies strictly above the critical curve of a related branching process, in the relevant parameter region. We also prove limit theorems for the percolation cluster in the supercritical, subcritical and critical regimes.

math.PR↗

arXiv2021

Asymptotic Results of a Multiple-entry Reinforcement Process

We introduce a class of stochastic processes with reinforcement consisting of a sequence of random partitions $\{\mathcal{P}_t\}_{t \ge 1}$, where $\mathcal{P}_t$ is a partition of $\{1,2,\dots, Rt\}$. At each time~$t$,~$R$ numbers are added to the set being partitioned; of these, a random subset (chosen according to a time-dependent probability distribution) joins existing blocks, and the others each start new blocks on their own. Those joining existing blocks each choose a block with probability proportional to that block's cardinality, independently. We prove results concerning the asymptotic cardinality of a given block and central limit theorems for associated fluctuations about this asymptotic cardinality: these are proved both for a fixed block and for the maximum among all blocks. We also prove that with probability one, a single block eventually takes and maintains the leadership in cardinality. Depending on the way one sees this partition process, one can translate our results to Balls and Bins processes, Generalized Chinese Restaurant Processes, Generalized Urn models and Preferential attachment random graphs.

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