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$1/3$
Let 
${\rm\Gamma}(n,p)$ denote the binomial model of a random triangular group. We show that there exist constants 
$c,C>0$ such that if 
$p\leqslant c/n^{2}$, then asymptotically almost surely (a.a.s.) 
${\rm\Gamma}(n,p)$ is free, and if 
$p\geqslant C\log n/n^{2}$, then a.a.s. 
${\rm\Gamma}(n,p)$ has Kazhdan’s property (T). Furthermore, we show that there exist constants 
$C^{\prime },c^{\prime }>0$ such that if 
$C^{\prime }/n^{2}\leqslant p\leqslant c^{\prime }\log n/n^{2}$, then a.a.s. 
${\rm\Gamma}(n,p)$ is neither free nor has Kazhdan’s property (T).
