arXiv subjects

Luiz Renato Fontes

Publications and source records attributed to Luiz Renato Fontes.

28 records · Page 2Linked to original sources

arXiv2015

Layered systems at the mean field critical temperature

We consider the Ising model on $\mathbb Z\times \mathbb Z$ where on each horizontal line $\{(x,i), x\in \mathbb Z\}$, the interaction is given by a ferromagnetic Kac potential with coupling strength $J_γ(x,y)\sim γJ(γ(x-y))$ at the mean field critical temperature. We then add a nearest neighbor ferromagnetic vertical interaction of strength $ε$ and prove that for every $ε>0$ the systems exhibits phase transition provided $γ>0$ is small enough.

math.PR↗

arXiv2014

GREM-like K processes on trees with infinite depth

We take up the issue of deriving the limit as $n\to\infty$ of the GREM-like K process on a tree with $n$ levels. Under specific conditions on the parameters of the process, implying the martingality of a modification of the underlying clock process sequences, we obtain infinite level clock processes as nontrivial limits of the finite level clocks, and use them to construct a process on a suitable product space which is then shown to be the limit of the $n$ level K processes as $n\to\infty$. Some properties of the limiting, infinite level K process are established, like an expression for the asymptotic empirical measure of cylinders, giving information on the prospective equilibrium measure of the infinite level dynamics.

math.PR↗

arXiv2014

Phase transitions in layered systems

We consider the Ising model on the two-dimensional square lattice where on each horizontal line, called "layer", the interaction is given by a ferromagnetic Kac potential with coupling strength $J_γ(x,y)=γJ(γ(x-y))$, where $J(\cdot)$ is smooth and has compact support; we then add a nearest neighbor ferromagnetic vertical interaction of strength $γ^{A}$ (where $A\ge 2$ is fixed) and prove that for any $β$ (inverse temperature) larger than the mean field critical value there is a phase transition for all $γ$ small enough.

math.PR↗

arXiv2014

Scaling limit of the radial Poissonian web

We consider a variant of the radial spanning tree introduced by Baccelli and Bordenave. Like the original model, our model is a tree rooted at the origin, built on the realization of a planar Poisson point process. Unlike it, the paths of our model have independent jumps. We show that locally our diffusively rescaled tree, seen as the collection of the paths connecting its sites to the root, converges in distribution to the Brownian Bridge Web, which is roughly speaking a collection of coalescing Brownian bridges starting from all the points of a planar strip perpendicular to the time axis, and ending at the origin.

math.PR↗

arXiv2013

Elementary results on K processes with weights

We introduce the title process via a particular construction, and relate it to processes previously studied, in particular a process introduced by G. E. H. Reuter in 1969. We derive elementary properties and quantities of this processes: Markov property, transition rates, stationary distribution, and the infinitesimal generator for a case not treated by Reuter.

math.PR↗

arXiv2013

On the dynamics of trap models in Z^d

We consider trap models on Z^d, namely continuous time Markov jump process on Z^d with embedded chain given by a generic discrete time random walk, and whose mean waiting time at x is given by tau_x, with tau = (tau_x, x in Z^d) a family of positive iid random variables in the basin of attraction of an alpha-stable law, 0<alpha<1. We may think of x as a trap, and tau_x as the depth of the trap at x. We are interested in the trap process, namely the process that associates to time t the depth of the currently visited trap. Our first result is the convergence of the law of that process under suitable scaling. The limit process is given by the jumps of a certain alpha-stable subordinator at the inverse of another alpha-stable subordinator, correlated with the first subordinator. For that result, the requirements for the embedded random walk are a) the validity of a law of large numbers for its range, and b) the slow variation at infinity of the tail of the distribution of its time of return to the origin: they include all transient random walks as well as all planar random walks, and also many one dimensional random walks. We then derive aging results for the process, namely scaling limits for some two-time correlation functions thereof, a strong form of which requires an assumption of transience, stronger than a, b. The above mentioned scaling limit result is an averaged result with respect to tau. Under an additional condition on the size of the intersection of the ranges of two independent copies of the embeddded random walk, roughly saying that it is small compared with the size of the range, we derive a stronger scaling limit result, roughly stating that it holds in probability with respect to tau. With that additional condition, we also strengthen the aging results, from the averaged version mentioned above, to convergence in probability with respect to tau.

math.PR↗

arXiv2007

Density-Profile Processes Describing Biological Signaling Networks: Almost Sure Convergence to Deterministic Trajectories

We introduce jump processes in R^k, called density-profile process, to model biological signaling networks. They describe the macroscopic evolution of finite-size spin-flip models with k types of spins interacting through a non-reversible Glauber dynamics. We focus on the the k-dimensional empirical-magnetization vector in the thermodynamic limit, and prove that, within arbitrary finite time-intervals, its path converges almost surely to a deterministic trajectory determined by a first-order (non-linear) differential equation. As parameters of the spin-flip dynamics change, the associated dynamical system may go through bifurcations, associated to phase transitions in the statistical mechanical setting. We present a simple example of spin-flip stochastic model leading to a dynamical system with Hopf and pitchfork bifurcations; depending on the parameter values, the magnetization random path can either converge to a unique stable fixed point, converge to one of a pair of stable fixed points, or asymptotically evolve close to a deterministic orbit in R^k.

math.PR↗

arXiv2006

K-processes, scaling limit and aging for the REM-like trap model

We study K-processes, which are Markov processes in a denumerable state space, all of whose elements are stable, with the exception of a single state, starting from which the process enters finite sets of stable states with uniform distribution. We show how these processes arise, in a particular instance, as scaling limits of the REM-like trap model ``at low temperature'', and subsequently derive aging results for those models in this context.

math.PR↗

arXiv2006

Threshold $theta geq 2$ contact processes on homogeneous trees

We study the threshold $theta geq 2$ contact process on a homogeneous tree $T_b$ of degree $kappa = b + 1$, with infection parameter $lambda geq 0$ and started from a product measure with density $p$. The corresponding mean-field model displays a discontinuous transition at a critical point $lambda_c^{MF}(kappa,theta)$ and for $lambda geq lambda_c^{MF}(kappa,theta)$ it survives iff $p geq p_c^{MF}(kappa,theta,lambda)$, where this critical density satisfies $0 < p_c^{MF}(kappa,theta,lambda) < 1$, $lim_{lambda to infty} p_c^{MF}(kappa,theta,lambda) = 0$. For large $b$, we show that the process on $T_b$ has a qualitatively similar behavior when $lambda$ is small, including the behavior at and close to the critical point $lambda_c(T_b,theta)$. In contrast, for large $lambda$ the behavior of the process on $T_b$ is qualitatively distinct from that of the mean-field model in that the critical density has $p_c(T_b,theta,infty) := lim_{lambda to infty} p_c(T_b,theta,lambda) > 0$. We also show that $lim_{b to infty} b lambda_c(T_b,theta) = Phi_{theta}$, where $1 < Phi_2 < Phi_3 < ...$, $lim_{theta to infty} Phi_{theta} = infty$, and $0 < liminf_{b to infty} b^{theta(theta-1)} p_c(T_b,theta,infty) leq limsup_{b to infty} b^{theta/(theta-1)} p_c(T_b,theta,infty) < infty$.

math.PR↗

arXiv2005

The Full Brownian Web as Scaling Limit of Stochastic Flows

In this paper we construct an object which we call the full Brownian web (FBW) and prove that the collection of all space-time trajectories of a class of one-dimensional stochastic flows converges weakly, under diffusive rescaling, to the FBW. The (forward) paths of the FBW include the coalescing Brownian motions of the ordinary Brownian web along with bifurcating paths. Convergence of rescaled stochastic flows to the FBW follows from general characterization and convergence theorems that we present here combined with earlier results of Piterbarg.

math.PR↗