## Final Report Summary - DSRELIS (Dynamics and Structure of Randomly Evolving Locally Interacting Systems (DSRELIS))

Randomly evolving systems consist of a large collection of small units that interact with one another over time giving rise to a dynamic large-scale structure. We focus on three systems: mobile point processes, allocation via sandpiles and random triangulations.

Mobile point processes. We consider a set of particles that are initially distributed over Z^2 as a Poisson point process of intensity \lambda, and move over Z^2 as independent nearest-neighbor random walks. Mobile point processes have been considered as prototype models for mobile networks, especially delay-tolerant networks such as vehicular networks, in which nodes moving in space cooperate to relay packets on behalf of others. One of our goals is to study the problem of target detection, which is a fundamental problem in the context of area surveillance by mobile networks. In this problem there is a target that is initially located at the origin and can move strategically to avoid colliding with particles. When the target collides with a particle, we say that the target is detected. An important question we solved is whether the particles will detect the presence of the target. We also solved the problem of spread of information through such particles, and built a general framework (based on the construction of a multi-scale Lipschitz surface) that can be used to analyze questions in this model. Finally, we solved the problem of aggregation known as multi-particle diffusion limited aggregation, by showing that the aggregate grows with positive speed in dimensions two and higher.

Allocation via sandpiles. We consider the fundamental problem in computer architecture of allocating tasks to processors that are interconnected as the vertices of a graph. In a previous work, I introduced the following allocation scheme. Particles (representing tasks) arrive one at a time to uniformly random vertices (processors), and move throughout the graph until being finally allocated to a vertex; so each vertex piles up the particles allocated to it. When a particle arrives, it moves to the adjacent vertex with the smallest pile, provided that the pile is smaller than the pile of the vertex the particle is currently on. The particle keeps moving in this way until it reaches a vertex whose pile is not larger than any of its neighbors; at this time, the particle is allocated to that vertex and does not move anymore. Then the next particle arrives, and the process is repeated. We proved a very general result relating the size of the largest pile to the structure of the graph.

Random triangulations. Lattice triangulations (triangulations of the NxN square lattice) are fundamental discrete geometric objects that received much attention in combinatorics, computational geometry and algebraic geometry. To better understand the complex structure and dynamical properties of lattice triangulations, in a previous work we introduced a parameter \lambda and let each triangulation T be selected with probability proportional to \lambda^{|T|}, where |T| denotes the sum of the lengths of the edges in T. Then \lambda=1 corresponds to uniformly random triangulations, while \lambda<1 favors triangulations with short edges and \lambda>1 favors triangulations with long edges. The only known pragmatic way to generate a random triangulation is via a Markov chain (or Glauber dynamics). The properties of this Markov chain and of random triangulations change substantially as \lambda varies, and a phase transition is conjectured to occur at \lambda=1. We established this for n x k triangulations, where k is some fixed constant. We also derived a powerful technique (based on a Lyapunov function) to study lattice triangulations of any lattice polygon in the regime \lambda<1, and studied the related model of dyadic dissections (which are a tiling of a square by rectangles of equal area, instead of triangles of equal area). In this latter object, we are able to analyze the mixing time in the critical, \lambda=1 case.

Mobile point processes. We consider a set of particles that are initially distributed over Z^2 as a Poisson point process of intensity \lambda, and move over Z^2 as independent nearest-neighbor random walks. Mobile point processes have been considered as prototype models for mobile networks, especially delay-tolerant networks such as vehicular networks, in which nodes moving in space cooperate to relay packets on behalf of others. One of our goals is to study the problem of target detection, which is a fundamental problem in the context of area surveillance by mobile networks. In this problem there is a target that is initially located at the origin and can move strategically to avoid colliding with particles. When the target collides with a particle, we say that the target is detected. An important question we solved is whether the particles will detect the presence of the target. We also solved the problem of spread of information through such particles, and built a general framework (based on the construction of a multi-scale Lipschitz surface) that can be used to analyze questions in this model. Finally, we solved the problem of aggregation known as multi-particle diffusion limited aggregation, by showing that the aggregate grows with positive speed in dimensions two and higher.

Allocation via sandpiles. We consider the fundamental problem in computer architecture of allocating tasks to processors that are interconnected as the vertices of a graph. In a previous work, I introduced the following allocation scheme. Particles (representing tasks) arrive one at a time to uniformly random vertices (processors), and move throughout the graph until being finally allocated to a vertex; so each vertex piles up the particles allocated to it. When a particle arrives, it moves to the adjacent vertex with the smallest pile, provided that the pile is smaller than the pile of the vertex the particle is currently on. The particle keeps moving in this way until it reaches a vertex whose pile is not larger than any of its neighbors; at this time, the particle is allocated to that vertex and does not move anymore. Then the next particle arrives, and the process is repeated. We proved a very general result relating the size of the largest pile to the structure of the graph.

Random triangulations. Lattice triangulations (triangulations of the NxN square lattice) are fundamental discrete geometric objects that received much attention in combinatorics, computational geometry and algebraic geometry. To better understand the complex structure and dynamical properties of lattice triangulations, in a previous work we introduced a parameter \lambda and let each triangulation T be selected with probability proportional to \lambda^{|T|}, where |T| denotes the sum of the lengths of the edges in T. Then \lambda=1 corresponds to uniformly random triangulations, while \lambda<1 favors triangulations with short edges and \lambda>1 favors triangulations with long edges. The only known pragmatic way to generate a random triangulation is via a Markov chain (or Glauber dynamics). The properties of this Markov chain and of random triangulations change substantially as \lambda varies, and a phase transition is conjectured to occur at \lambda=1. We established this for n x k triangulations, where k is some fixed constant. We also derived a powerful technique (based on a Lyapunov function) to study lattice triangulations of any lattice polygon in the regime \lambda<1, and studied the related model of dyadic dissections (which are a tiling of a square by rectangles of equal area, instead of triangles of equal area). In this latter object, we are able to analyze the mixing time in the critical, \lambda=1 case.