The probability of majority rule instability in the 2D euclidean model with an even number of voters

The classic instability theorems of Euclidean voting theory definitively treat all cases except that of an even number of voters in two dimensions. For that case, all that has been known is that the set of stable configurations is neither measure 0 nor measure 1. We prove that instability occurs with probability converging rapidly to 1 as the population increases.