Zero-one laws for graphs with edge probabilities decaying with distance. Part II

Let G_nn be the random graph on [n] = {1, . . . , n} with the probability of {i, j} being an edge decaying as a power of the distance, specifically the probability being P|i-j| = 1/|i -j|^α, where the constant α ∈ (0, 1) is irrational. We analyze this theory using an appropriate weight function on a pair (A, B) of graphs and using an equivalence relation on B \ A. We then investigate the model theory of this theory, including a "finite compactness". Lastly, as a consequence, we prove that the zero-one law (for first order logic) holds.