Interacting Particle Systems, cellular automata, quasilocality and color representations

Research Project, 2021 – 2024

The first subproject will concern interacting particle systems and cellular automata. Here one wants to (1) understand the spatial and time-ergodic consequences of having a unique stationary distribution, (2) obtain new fixation results using sophisticated techniques from ergodic theory, (3) determining the asymptotic behavior of various fixation times, and (4) use such results to attack Liggett´s conjecture concerning the stochastic Ising model.The second subproject concerns obtaining new results concerning quasilocality, g-functions, Gibbs states and tail sigma-algebras and a better understanding on how all these relate.The third subproject is to see if and how the recent joint work with Malin Palö Forsström concerning color representations and their connections to threshold Gaussian and stable vectors can be used to obtain new insights into the structure of Gaussian fields.The fourth subproject concerns analyzing a certain problem involving optimization, a general distribution in n-space and a certain payoff function; the problem can beinterpreted as asking how far one should stand from the target when playing darts. It is difficult to predict how the timing will work out but we hope to obtain various results during the project period. Also, as always in mathematics, one cannot say which methods will turn out to be successful; one sees as one goes along. The planned reseach is important since, if successful, it would yield new, interesting and difficult results.