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Let $\alpha\in(0,1)_\mathbb{R}$ be irrational and $G_n = G_{{n, 1/n}^\alpha}$be the random graph with edge probability $1/n^\alpha$; we know that itsatisfies the 0-1 law for first order logic. We deal with the failure of the0-1 law for stronger logics: $\mathbb{L}_{ \infty, k}, k$ large enough and theLFP, least fix point logic.

Failure of 0-1 law for sparse random graph in strong logics