arXiv subjects

Daniel Valesin

Publications and source records attributed to Daniel Valesin.

43 records · Page 3Linked to original sources

arXiv2014

Phase transition of the contact process on random regular graphs

We consider the contact process with infection rate $λ$ on a random $(d+1)$-regular graph with $n$ vertices, $G_n$. We study the extinction time $τ_{G_n}$ (that is, the random amount of time until the infection disappears) as $n$ is taken to infinity. We establish a phase transition depending on whether $λ$ is smaller or larger than $λ_1(\mathbb{T}^d)$, the lower critical value for the contact process on the infinite, $(d+1)$-regular tree: if $λ< λ_1(\mathbb{T}^d)$, $τ_{G_n}$ grows logarithmically with $n$, while if $λ> λ_1(\mathbb{T}^d)$, it grows exponentially with $n$. This result differs from the situation where, instead of $G_n$, the contact process is considered on the $d$-ary tree of finite height, since in this case, the transition is known to happen instead at the _upper_ critical value for the contact process on $\mathbb{T}^d$.

math.PR↗

arXiv2014

The contact process on finite homogeneous trees revisited

We consider the contact process with infection rate $λ$ on $\mathbb{T}_n^d$, the $d$-ary tree of height $n$. We study the extinction time $τ_{\mathbb{T}_n^d}$, that is, the random time it takes for the infection to disappear when the process is started from full occupancy. We prove two conjectures of Stacey regarding $τ_{\mathbb{T}_n^d}$. Let $λ_2$ denote the upper critical value for the contact process on the infinite $d$-ary tree. First, if $λ< λ_2$, then $τ_{\mathbb{T}_n^d}$ divided by the height of the tree converges in probability, as $n \to \infty$, to a positive constant. Second, if $λ> λ_2$, then $\log \mathbb{E}[τ_{\mathbb{T}_n^d}]$ divided by the volume of the tree converges in probability to a positive constant, and $τ_{\mathbb{T}_n^d}/\mathbb{E}[τ_{\mathbb{T}_n^d}]$ converges in distribution to the exponential distribution of mean 1.

math.PR↗

arXiv2012

Metastable Densities for Contact Processes on Power Law Random Graphs

We consider the contact process on a random graph with fixed degree distribution given by a power law. We follow the work of Chatterjee and Durrett, who showed that for arbitrarily small infection parameter $λ$, the survival time of the process is larger than a stretched exponential function of the number of vertices, $n$. We obtain sharp bounds for the typical density of infected sites in the graph, as $λ$ is kept fixed and $n$ tends to infinity. We exhibit three different regimes for this density, depending on the tail of the degree law.

math.PR↗

arXiv2012

Supercriticality of Annealed Approximations of Boolean Networks

We consider a model recently proposed by Chatterjee and Durrett [CD2011] as an "annealed approximation" of boolean networks, which are a class of cellular automata on a random graph, as defined by S. Kauffman [K69]. The starting point is a random directed graph on $n$ vertices; each vertex has $r$ input vertices pointing to it. For the model of [CD2011], a discrete time threshold contact process is then considered on this graph: at each instant, each vertex has probability $q$ of choosing to receive input; if it does, and if at least one of its input vertices were in state 1 at the previous instant, then it is labelled with a 1; in all other cases, it is labelled with a 0. $r$ and $q$ are kept fixed and $n$ is taken to infinity. Improving a result of [CD2011], we show that if $qr > 1$, then the time of persistence of activity of the dynamics is exponential in $n$.

math.PR↗

arXiv2012

Exponential extinction time of the contact process on finite graphs

We study the extinction time $\uptau$ of the contact process on finite trees of bounded degree. We show that, if the infection rate is larger than the critical rate for the contact process on $\Z$, then, uniformly over all trees of degree bounded by a given number, the expectation of $\uptau$ grows exponentially with the number of vertices. Additionally, for any sequence of growing trees of bounded degree, $\uptau$ divided by its expectation converges in distribution to the unitary exponential distribution. These also hold if one considers a sequence of graphs having spanning trees with uniformly bounded degree. Using these results, we consider the contact process on a random graph with vertex degrees following a power law. Improving a result of Chatterjee and Durrett \cite{CD}, we show that, for any infection rate, the extinction time for the contact process on this graph grows exponentially with the number of vertices.

math.PR↗

arXiv2010

Tightness for the interface of the one-dimensional contact process

We consider a symmetric, finite-range contact process with two types of infection; both have the same (supercritical) infection rate and heal at rate 1, but sites infected by Infection 1 are immune to Infection 2. We take the initial configuration where sites in $(-\infty,0]$ have Infection 1 and sites in $[1,\infty)$ have Infection 2, then consider the process $ρ_t$ defined as the size of the interface area between the two infections at time $t$. We show that the distribution of $ρ_t$ is tight, thus proving a conjecture posed by Cox and Durrett in [Bernoulli 1 (1995) 343--370].

math.PR↗

arXiv2010

Multitype Contact Process on $\Z$: Extinction and Interface

We consider a two-type contact process on $\Z$ in which both types have equal finite range and supercritical infection rate. We show that a given type becomes extinct with probability 1 if and only if, in the initial configuration, it is confined to a finite interval $[-L,L]$ and the other type occupies infinitely many sites both in $(-\infty, L)$ and $(L, \infty)$. We also show that, starting from the configuration in which all sites in $(-\infty, 0]$ are occupied by type 1 particles and all sites in $(0, \infty)$ are occupied by type 2 particles, the process $ρ_t$ defined by the size of the interface area between the two types at time $t$ is tight.

math.PR↗