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* ACTIVATED RANDOM WALK MODEL * This is a conservative particle system on thelattice, with a Markovian continuous-time evolution. Active particles performrandom walks without interaction, and they may as well change their state topassive, then stopping to jump. When particles of both types occupy the samesite, they all become active. This model exhibits phase transition in the sensethat for low initial densities the system locally fixates and for highdensities it keeps active. Though extensively studied in the physicsliterature, the matter of giving a mathematical proof of such phase transitionremained as an open problem for several years. In this work we identify somevariables that are sufficient to characterize fixation and at the same time arestochastically monotone in the model's parameters. We employ an explicitgraphical representation in order to obtain the monotonicity. With this methodwe prove that there is a unique phase transition for the one-dimensionalfinite-range random walk. Joint with V. Sidoravicius.  * BROKEN LINE PROCESS * We introduce the broken line process and derive someof its properties. Its discrete version is presented first and a naturalgeneralization to the continuum is then proposed and studied. The broken linesare related to the Young diagram and the Hammersley process and are useful forcomputing last passage percolation values and finding maximal oriented paths.For a class of passage time distributions there is a family of boundaryconditions that make the process stationary and reversible. One application isa simple proof of the explicit law of large numbers for last passagepercolation with exponential and geometric distributions. Joint with V.Sidoravicius, D. Surgailis, and M. E. Vares.

Generalized Hammersley Process and Phase Transition for Activated Random Walk Models