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We consider the Brownian SYK model of $N$ interacting Majorana fermions, withrandom couplings that are taken to vary independently at each time. We studythe out-of-time-ordered correlators (OTOCs) of arbitrary observables and theR\'enyi-$2$ tripartite information of the unitary evolution operator, whichwere proposed as diagnostic tools for quantum chaos and scrambling,respectively. We show that their averaged dynamics can be studied as a quenchproblem at imaginary times in a model of $N$ qudits, where the Hamiltoniandisplays site-permutational symmetry. By exploiting a description in terms ofbosonic collective modes, we show that for the quantities of interest thedynamics takes place in a subspace of the effective Hilbert space whosedimension grows either linearly or quadratically with $N$, allowing us toperform numerically exact calculations up to $N = 10^6$. We analyze in detailthe interesting features of the OTOCs, including their dependence on the chosenobservables, and of the tripartite information. We observe explicitly theemergence of a scrambling time $t^\ast\sim \ln N$ controlling the onset of bothchaotic and scrambling behavior, after which we characterize the exponentialdecay of the quantities of interest to the corresponding Haar scrambled values.

Quantum chaos in the Brownian SYK model with large finite $N$: OTOCs and tripartite information