Finding a non-sofic hyperbolic group will resolve two major problems in geometric group theory: Are there non sofic groups? Are there non residually finite hyperbolic groups? In this paper, we propose a new probabilistic approach to this problem, based on the cocycle stability in permutations of random 2-dimensional Linial-Meshulam complexes. Specifically, we study their cocycle stability rate, which measures how far cochains with small coboundaries are from being cocycles. Our main contribution is the following: If, in a middle triangle density range, these random complexes typically have a linear cocycle stability rate, then there exists a non-sofic hyperbolic group. Our proof method is inspired by a well known fact about the non local testability of Sipser-Spielman expander codes.