Open problems

These problems are mostly original, and (of course) reflect my personal taste. But I have tried to sample different topics.

I categorize problems on a conceptual-technical spectrum. That is

Type 1 We have a question in words, but we don't know how to state a precise mathematical problem which both seems capable of solution and whose solution would be interesting.
Type 2 We have a precise mathematical problem, but we do not see any plausible outline for a potential proof.
Type 3 We have a precise mathematical problem, and a plausible outline for a potential proof, but we cannot carry through the technical details.

Of course these overlap. Continuize by taking "type 2" in the middle of an inteval 1.5 - 2.5. Then we can order the problems across this spectrum.

(0.5) Martingale, for practical purposes.
(0.7) Congestion in networks: SOC, HOT and the percolation-fragmentation phase transition.
(1.7) The constrained Ising model as an algorithm for storage in dynamic graphs.
(1.8) Bacon and eggs: task scheduling into batches.
(1.9) Metropolis on Cayley graphs.
(2.0) Coding invariant processes on infinite trees.
(2.1) Mixing times for the branch-rotation chain on cladograms.
(2.2) Scaling window for percolation of averages in the mean-field setting.
(2.6) Percolation and empires.
(2.8) Power laws and killed branching random walk.
(3.0) Wright-Fisher diffusions with negative mutation rate!
(3.2) How many Brownian particles escape when you control with total drift = 1?.
(3.3) The low-density limit of coalescing branching random walk.

In a different realm, I really really really want someone to do this simulation. Of course, it's not easy!

Former open problems, now solved