  We consider exact asymptotics of the minimax risk for global testing against
sparse alternatives in the context of high dimensional linear regression. Our
results characterize the leading order behavior of this minimax risk in several
regimes, uncovering new phase transitions in its behavior. This complements a
vast literature characterizing asymptotic consistency in this problem, and
provides a useful benchmark, against which the performance of specific tests
may be compared. Finally, we provide some preliminary evidence that popular
sparsity adaptive procedures might be sub-optimal in terms of the minimax risk.
