Alignment
Oriented particles, following a local stochastic rule, must align their orientations.
Problem statement and measuresexpand
Each particle carries one of \(q \ge 2\) orientations, a spin state, not a sense of direction. Particles run a local Markov chain; the system should be aligned (most particles share an orientation) or not, depending on the bias, with high probability in the stationary distribution, both when connectivity is kept and when particles may disconnect.
Measures: Time, asynchronous rounds; Moves, total expansions and contractions (work).
Progress map
tight near-optimal polynomial gap unknown impossible· rows follow the assumption tree; each cell: best upper bound vs lower bound, and what forces the lower bound
Setting
Time
Moves
unknown
\(\text{eventually}\) vs —
unknown
— vs —
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