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Activity transition?Start on an infinite lattice, with a Poisson(rho) number of particles at every lattice site. Then we topple in parallel: from every site that has at least two particles, two particles go to a neighbor, each choosing with equal probability the right or the bottom neighbor. The movie shows this model on a 100x100 grid with periodic boundary conditions, and rho = 0.51 (restart every 500 frames). Colors: zero particles = dark blue, one particle is light blue, etc. Open problem: for which rho does the activity die out almost surely? 


Limiting shape?Start on an infinite lattice, with height n=4 at the origin and h=47/64 at every other lattice site. Then start splitting: every site with height at least 1, distributes its heigth evenly among its neighbors, so that its own height becomes 0. This is a growth model: the set of sites that splits increases in time. The movie shows the first 50 time steps. But could you have predicted that it grows like an octagon? We proved three different shapes for as many intervals of h, but many others, including this one, are still open. 
Fractal pattern?Start on an infinite empty lattice. We add n particles to the origin, and let them run off: every lattice site that has at least two particles, gives one to its right neighbor and one to its bottom neighbor. Eventually, every site has zero particles (dark blue) or one particle (light blue). The movie shows this model with n = 800, added 100 at the time. Open problem: what can we say about the emerging triangle pattern as n tends to infinity? 


'temperature' profile?There is an interest in interacting particle systems for modeling thermal nonequilibrium. How about this one: On the line Z, fix the height of the origin to 100, and start with all other sites at height 0. Then in every time step, topple each site with a probability proportional to its height. In a toppling, the height of a site decreases by 2 and that of the neighboring sites increases by 1. What can we say about the resulting height profile, and fluctuations? 