One of the famous outstanding open problems in probability theory (discussed e.g. in this MathOverflow post) is the following question:
Consider the points with integer coordinates in 3D, and join adjacent points with an edge independently with probability p. (This is "bond percolation on the 3D integer lattice".) It is a basic result of percolation theory that there is a "critical probability" p_c such that for p < p_c, there is no infinite connected component (with probability 1) and for p > p_c, there is an infinite connected component (with probability 1).
The open question is whether or not there is an infinite connected component (with probability 1) at the exact critical probability p = p_c.
Resolves YES if, by the end of 2035, there is general consensus among the mathematical community that there is a rigorous proof establishing an answer (either in the positive or negative) to this question. Resolves NO otherwise.
(I will not trade in this market.)
ADDITIONAL CONTEXT:
- it is known that the probability of there being an infinite cluster is either 0 or 1;
- it is known that there is no infinite cluster in dimension 2, and also in dimensions d >= 19.
1,000