  * ACTIVATED RANDOM WALK MODEL * This is a conservative particle system on the
lattice, with a Markovian continuous-time evolution. Active particles perform
random walks without interaction, and they may as well change their state to
passive, then stopping to jump. When particles of both types occupy the same
site, they all become active. This model exhibits phase transition in the sense
that for low initial densities the system locally fixates and for high
densities it keeps active. Though extensively studied in the physics
literature, the matter of giving a mathematical proof of such phase transition
remained as an open problem for several years. In this work we identify some
variables that are sufficient to characterize fixation and at the same time are
stochastically monotone in the model's parameters. We employ an explicit
graphical representation in order to obtain the monotonicity. With this method
we prove that there is a unique phase transition for the one-dimensional
finite-range random walk. Joint with V. Sidoravicius.
  * BROKEN LINE PROCESS * We introduce the broken line process and derive some
of its properties. Its discrete version is presented first and a natural
generalization to the continuum is then proposed and studied. The broken lines
are related to the Young diagram and the Hammersley process and are useful for
computing last passage percolation values and finding maximal oriented paths.
For a class of passage time distributions there is a family of boundary
conditions that make the process stationary and reversible. One application is
a simple proof of the explicit law of large numbers for last passage
percolation with exponential and geometric distributions. Joint with V.
Sidoravicius, D. Surgailis, and M. E. Vares.
