  The Ising $p$-spin glass and the random $k$-SAT models exhibit symmetric
multi Overlap Gap Property ($m$-OGP), an intricate geometrical property which
is a rigorous barrier against many important classes of algorithms. We
establish that for both models, the symmetric $m$-OGP undergoes a sharp phase
transition and we identify the phase transition point for each model: for any
$m\in\mathbb{N}$, there exists $\gamma_m$ (depending on the model) such that
the model exhibits $m$-OGP for $\gamma>\gamma_m$ and the $m$-OGP is provably
absent for $\gamma<\gamma_m$, both with high probability, where $\gamma$ is
some natural parameter of the model. Our results for the Ising $p$-spin glass
model are valid for all large enough $p$ that remain constant as the number $n$
of spins grows, $p=O(1)$; our results for the random $k$-SAT are valid for all
$k$ that grow mildly in the number $n$ of Boolean variables, $k=\Omega(\ln n)$.
To the best of our knowledge, these are the first sharp phase transition
results regarding the $m$-OGP. Our proofs are based on an application of the
second moment method combined with a concentration property regarding a
suitable random variable. While a standard application of the second moment
method fails, we circumvent this issue by leveraging an elegant argument of
Frieze~\cite{frieze1990independence} together with concentration.
