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Remy Sanchis

Publications and source records attributed to Remy Sanchis.

At least 19 recordsLinked to original sources
arXiv2025

Phase transition on randomly horizontally stretched square lattice

In this article, we study a bond percolation model on a horizontally stretched square lattice, constructed by stretching the distances between the columns of $\mathbb{Z}_+^2$ according to a collection of independent and identically distributed (i.i.d.) copies of a non-negative random variable $\xi$. We assume that $\xi$ satisfies the integrability condition \[ \mathbb{E}\big[\xi\, e^{c(\log \xi)^{1/2}} \,\mathbb{1}_{\{\xi \geq 1\}}\big] < \infty, \] for some constant $c > 8\sqrt{\log 96}$. In this random environment, each vertical edge is independently declared open with probability $p$, while each horizontal edge is open with probability $p^{|e|}$, where $|e|$ denotes the Euclidean length of the edge. We develop a multiscale renormalization scheme adapted to this geometry and use it to prove that percolation occurs for all sufficiently large values of $p < 1$.

math.PR↗
arXiv2023

A new proof for percolation phase transition on stretched lattices

We revisit the phase transition for percolation on randomly stretched lattices. Starting with the usual square grid, keep all vertices untouched while erasing edges according as follows: for every integer $i$, the entire column of vertical edges contained in the line $\{ x = i \}$ is removed independently of other columns with probability $ρ> 0$. Similarly, for every integer $j$, the entire row of horizontal edges contained in the line $\{ y = j\}$ is removed independently with probability $ρ$. On the remaining random lattice, we perform Bernoulli bond percolation. Our main contribution is an alternative proof that the model undergoes a nontrivial phase transition, a result established earlier by Hoffman. The main novelty lies on the fact that the dynamic renormalization employed earlier is replaced by a static version, which is simpler and more robust to extend to different models. We emphasize the flexibility of our methods by showing the non-triviality of the phase transition for a new oriented percolation model in a random environment as well as for a model previously investigated by Kesten, Sidoravicius and Vares. We also prove a result about the sensitivity of the phase transition with respect to the stretching mechanism.

math.PR↗
arXiv2023

Diameter of P.A. random graphs with edge-step functions

In this work we prove general bounds for the diameter of random graphs generated by a preferential attachment model whose parameter is a function $f:\mathbb{N}\to[0,1]$ that drives the asymptotic proportion between the numbers of vertices and edges. These results are sharp when $f$ is a \textit{regularly varying function at infinity} with strictly negative index of regular variation~$-γ$. For this particular class, we prove a characterization for the diameter that depends only on~$-γ$. More specifically, we prove that the diameter of such graphs is of order $1/γ$ with high probability, although its vertex set order goes to infinity polynomially. Sharp results for the diameter for a wide class of \textit{slowly varying functions} are also obtained.

math.PR↗
arXiv2023

Anisotropic non-oriented bond percolation in high dimensions

We consider inhomogeneous non-oriented Bernoulli bond percolation on $\mathbb{Z}^d$, where each edge has a parameter depending on its direction. We prove that, under certain conditions, if the sum of the parameters is strictly greater than 1/2, we have percolation in sufficiently high dimensions. The main tool is a dynamical coupling between models for different dimensions with different sets of parameters.

math.PR↗
arXiv2021

Anisotropic oriented percolation in high dimensions

In this paper we study anisotropic oriented percolation on $\mathbb{Z}^d$ for $d\geq 4$ and show that the local condition for phase transition is closely related to the mean-field condition. More precisely, we show that if the sum of the local probabilities is strictly greater than one and each probability is not too large, then percolation occurs.

math.PR↗
arXiv2020

A note on the dimensional crossover critical exponent

We consider independent anisotropic bond percolation on $\mathbb{Z}^d\times \mathbb{Z}^s$ where edges parallel to $\mathbb{Z}^d$ are open with probability $p<p_c(\mathbb{Z}^d)$ and edges parallel to $\mathbb{Z}^s$ are open with probability $q$, independently of all others. We prove that percolation occurs for $q\geq 8d^2(p_c(\mathbb{Z}^d)-p)$. This fact implies that the so-called Dimensional Crossover critical exponent, if it exists, is greater than 1. In particular, using known results, we conclude the proof that, for $d\geq 11$, the crossover critical exponent exists and equals 1.

math.PR↗
arXiv2020

Phase transition for percolation on a randomly stretched lattice

Let $\{ξ_i\}_{i \geq 1}$ be a sequence of i.i.d.\ positive random variables. Starting from the usual square lattice replace each horizontal edge that links a site in $i$-th vertical column to another in the $(i+1)$-th vertical column by an edge having length $ξ_i$. Then declare independently each edge $e$ in the resulting lattice open with probability $p_e=p^{|e|}$ where $p\in[0,1]$ and $|e|$ is the length of $e$. We relate the occurrence of nontrivial phase transition for this model to moment properties of $ξ_1$. More precisely, we prove that the model undergoes a nontrivial phase transition when $\mathbb{E}(ξ_1^η)<\infty$, for some $η>1$ whereas, when $\mathbb{E}(ξ_1^η)=\infty$ for some $η<1$, no phase transition occurs.

math.PR↗
arXiv2019

A local lemma via entropy compression

In the framework of the probabilistic method in combinatorics, we revisit the entropy compression method clarifying the setting in which it can be applied and providing a theorem yielding a general constructive criterion. We finally elucidate, through topical examples, the effectiveness of the entropy-compression criterion in comparison with the Lovasz Local Lemma criterion and, in particular, with the improved criterion based on cluster expansion.

math.CO↗
arXiv2019

Contact Process under heavy-tailed renewals on finite graphs

We investigate a non-Markovian analogue of the Harris contact process in a finite connected graph G=(V,E): an individual is attached to each site x in V, and it can be infected or healthy; the infection propagates to healthy neighbors just as in the usual contact process, according to independent exponential times with a fixed rate lambda>0; however, the recovery times for an individual are given by the points of a renewal process attached to its timeline, whose waiting times have distribution mu such that mu(t,infty) = t^{-alpha}L(t), where 1/2 < alpha < 1 and L is a slowly varying function; the renewal processes are assumed to be independent for different sites. We show that, starting with a single infected individual, if |V| < 2 + (2 alpha -1)/[(1-alpha)(2-alpha)], then the infection does not survive for any lambda; and if |V| > 1/(1-alpha), then, for every lambda, the infection has positive probability to survive

math.PR↗
arXiv2019

Preferential Attachment Random Graphs with Edge-Step Functions

We propose a random graph model with preferential attachment rule and \emph{edge-step functions} that govern the growth rate of the vertex set. We study the effect of these functions on the empirical degree distribution of these random graphs. More specifically, we prove that when the edge-step function $f$ is a \emph{monotone regularly varying function} at infinity, the sequence of graphs associated to it obeys a power-law degree distribution whose exponent is related to the index of regular variation of $f$ at infinity whenever said index is greater than $-1$. When the regularly variation index is less than or equal to $-1$, we show that the proportion of vertices with degree smaller than any given constant goes to $0$ a. s..

math.PR↗
arXiv2019

Agglomeration in a preferential attachment random graph with edge-steps

In this paper we investigate geometric properties of graphs generated by a preferential attachment random graph model with edge-steps. More precisely, at each time $t\in\mathbb{N}$, with probability $p$ a new vertex is added to the graph (a vertex-step occurs) or with probability $1-p$ an edge connecting two existent vertices is added (an edge-step occurs). We prove that the global clustering coefficient decays as $t^{-γ(p)}$ for a positive function $γ$ of $p$. We also prove that the clique number of these graphs is, up to sub-polynomially small factors, of order~$t^{(1-p)/(2-p)}$.

math.PR↗
arXiv2016

Disparity of clustering coefficients in the Holme-Kim network model

The Holme-Kim random graph processes is a variant of the Barabasi-Albert scale-free graph that was designed to exhibit clustering. In this paper we show that whether the model does indeed exhibit clustering depends on how we define the clustering coefficient. In fact, we find that local clustering coefficient remains typically positive whereas global clustering tends to 0 at a slow rate. These and other results are proven via martingale techniques, such as Freedman's concentration inequality combined with a bootstrapping argument.

math.PR↗
arXiv2016

Large Communities in a scale-free network

We prove the existence of a large complete subgraph w.h.p. in a preferential attachment random graph process with an edge-step. That is, we prove that the random graph $G_{t}$ produced by the so-called GLP model at time $t$ contains a complete subgraph of order $t^α$, where $α= (1-\varepsilon)\frac{1-p}{2-p}$, $\varepsilon$ is any number such that $0<\varepsilon<1$, and $0<p<1$ is a parameter of the model.

math.PR↗
arXiv2012

Percolation on infinite graphs and isoperimetric inequalities

We consider the Bernoulli bond percolation process (with parameter $p$) on infinite graphs and we give a general criterion for bounded degree graphs to exhibit a non-trivial percolation threshold based either on a single isoperimetric inequality if the graph has a bi-infinite geodesic, or two isoperimetric inequalities if the graph has not a bi-infinite geodesic. This new criterion extends previous criteria and brings together a large class of amenable graphs (such as regular lattices) and non-amenable graphs (such trees). We also study the finite connectivity in graphs satisfying the new general criterion and show that graphs in this class with a bi-infinite geodesic always have finite connectivity functions with exponential decay as $p$ is sufficiently close to one. On the other hand, we show that there are graphs in the same class with no bi-infinite geodesic for which the finite connectivity decays sub-exponentially (down to polynomially) in the highly supercritical phase even for $p$ arbitrarily close to one.

math-ph↗
arXiv2009

Percolation of words on $\Z^d$ with long range connections

Consider an independent site percolation model on $\Z^d$, with parameter $p \in (0,1)$, where all long range connections in the axes directions are allowed. In this work we show that given any parameter $p$, there exists and integer $K(p)$ such that all binary sequences (words) $ξ\in \{0,1\}^{\N}$ can be seen simultaneously, almost surely, even if all connections whose length is bigger than $K(p)$ are suppressed. We also show some results concerning the question how $K(p)$ should scale with $p$ when $p$ goes to zero. Related results are also obtained for the question of whether or not almost all words are seen.

math.PR↗