Generalized Hammersley Process and Phase Transition For Activated Random Walk Models
Broken Line Process | Activated Random Walk Model | Phase Transition
This thesis consists of two parts that are independent of each other. In the first part I report on a joint work with V.~Sidoravicius concerning the 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 monotonicity. This representation has a very useful commutativity property that allows direct, constructive approaches. With this method we prove that there is a unique phase transition for the one-dimensional finite-range random walk. In the second part I report on a joint work with V.~Sidoravicius, D.~Surgailis, and M.~E.~Vares about the Broken Line Process. In this work 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. For such distributions there is a law of large numbers and the process extends to the infinite lattice. One application is a simple proof of the explicit law of large numbers for last passage percolation with exponential and geometric distributions.