# Noise sensitivity, optimal graphs and perturbations of interacting particle systems

This proposed project has three subprojects. Some parts of the project are suitable for a Ph.D. student. The first part deals with noise sensitivity. One has n coin flips and some function f of these. One perturbs the coins by switching each coin with a small probability. How much does this affect the output f? (1) very little or (2) very much? The spectral behavior of f is critical here where (1) corresponds to ´low spectrum´ and (2) corresponds to ´high spectrum´. Benjamini, Kalai and Schramm proved (2) holds for crossings in percolation. Quantitative refinements of their work were done first by Schramm and Steif and then by Garban, Pete and Schramm. The first part of the project is to extend our knowledge of noise sensitivity. Some examples of directions to investigate are (1) the relationship between the spectral sample and the pivotal set, (2) the relationship between the Fourier spectrum and issues related to randomized algorithms and (3) noise sensitivity under other types of dynamics. In the second subproject, one wants to study particle systems (PS) on finite graphs and understand which properties of the graph are responsible for various aspects of the evolution, such as the behavior of absorption times. Which graphs yield the largest (smallest) expected absorption time? In the third subproject, the goal is to study PS on infinite graphs and understand how local changes affect the global behavior of the system, such as the set of stationary distributions.

### Funded by

- Swedish Research Council (VR) (Public, Sweden)