  We analyze the out-of-time-order correlation functions of a solvable model of
a large number, $N$, of $M$-component quantum rotors coupled by
Gaussian-distributed random, infinite-range exchange interactions. We focus on
the growth of commutators of operators at a temperature $T$ above the zero
temperature quantum critical point separating the spin-glass and paramagnetic
phases. In the large $N,~M$ limit, the squared commutators of the rotor fields
do not display any exponential growth of commutators, in spite of the absence
of any sharp quasiparticle-like excitations in the disorder-averaged theory. We
show that in this limit, the problem is integrable and point out interesting
connections to random-matrix theory. At leading order in $1/M$, there are no
modifications to the critical behavior but an irrelevant term in the
fixed-point action leads to a small exponential growth of the squared
commutator. We also introduce and comment on a generalized model involving
$p$-pair rotor interactions.
