Generalized Hammersley Process and Phase Transition for Activated Random Walk Models

Details Generalized Hammersley Process and Phase Transition for Activated Random Walk Models Generalized Hammersley Process and Phase Transition for Activated Random Walk Models

2008-12-14

arxiv.org/abs/0812.2473v1

Mathematics

Probability

PhD thesis. 89 pages, 6 figues. LaTeX

Scientific paper

* 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.

Affiliated with

Also associated with

No associations

Rating

Generalized Hammersley Process and Phase Transition for Activated Random Walk Models does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Generalized Hammersley Process and Phase Transition for Activated Random Walk Models, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Generalized Hammersley Process and Phase Transition for Activated Random Walk Models will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-249853