Leandro P. R. Pimentel

Asymptotics for first-passage times on delaunay triangulations

In this paper we study planar first-passage percolation (FPP) models on random Delaunay triangulations. In [14], Vahidi-Asl and Wierman showed, using sub-additivity theory, that the rescaled first-passage time converges to a finite and non-negative constant μ. We show a sufficient condition to ensure that μ>0 and derive some upper bounds for fluctuations. Our proofs are based on percolation ideas and on the method of martingales with bounded increments.

Busemann functions and the speed of a second class particle in the rarefaction fan

In this paper we will show how the results found in [Probab. Theory Related Fields 154 (2012) 89–125], about the Busemann functions in last-passage percolation, can be used to calculate the asymptotic distribution of the speed of a single second class particle starting from an arbitrary deterministic configuration which has a rarefaction fan, in either the totally asymetric exclusion process or the Hammersley interacting particle process. The method will be to use the well-known last-passage percolation description of the exclusion process and of the Hammersley process, and then the well-known connection between second class particles and competition interfaces.

Duality between coalescence times and exit points in last-passage percolation models

In this paper we prove a duality relation between coalescence times and exit points in last-passage percolation models with exponential weights. As a consequence, we get lower bounds for coalescence times with scaling exponent 3/2, and we relate its distribution with variational problems involving the Brownian motion process and the Airy process.

Local Behaviour of Airy Processes

The Airy processes describe limit fluctuations in a wide range of growth models, where each particular Airy process depends on the geometry of the initial profile. We show how the coupling method, developed in the last-passage percolation context, can be used to prove existence of a continuous version and local convergence to Brownian motion. By using similar arguments, we further extend these results to a two parameter limit fluctuation process (Airy sheet).

On the location of the maximum of a process: Lévy, Gaussian and Random field cases

ABSTRACT We show how the techniques presented in Pimentel [On the location of the maximum of a continuous stochastic process, J. Appl. Prob. 51 (2014), pp. 152–161] can be extended to a variety of non-continuous processes and random fields. For the Gaussian case, we prove new covariance formulae between the maximum and the maximizer of the process. As examples, we prove uniqueness of the location of the maximum for spectrally positive Lévy processes, Ornstein–Uhlenbeck process, fractional Brownian Motion and the Brownian sheet among other processes.

Shock Fluctuations for the Hammersley Process

We consider the Hammersley interacting particle system starting from a shock initial profile with densities

On the location of the maximum of a continuous stochastic process

\rho ,\lambda \in {\mathbb R}

A shape theorem and semi-infinite geodesics for the Hammersley model with random weights

ρ,λ∈R (