  Several asymptotic results for the implied volatility generated by a rough
volatility model have been obtained in recent years (notably in the
small-maturity regime), providing a better understanding of the shapes of the
volatility surface induced by rough volatility models, and supporting their
calibration power to S&P500 option data. Rough volatility models also generate
a local volatility surface, via the so-called Markovian projection of the
stochastic volatility. We complement the existing results on the implied
volatility by studying the asymptotic behavior of the local volatility surface
generated by a class of rough stochastic volatility models, encompassing the
rough Bergomi model. Notably, we observe that the celebrated "1/2 skew rule"
linking the short-term at-the-money skew of the implied volatility to the
short-term at-the-money skew of the local volatility, a consequence of the
celebrated "harmonic mean formula" of [Berestycki, Busca, and Florent, QF
2002], is replaced by a new rule: the ratio of the at-the-money implied and
local volatility skews tends to the constant 1/(H + 3/2) (as opposed to the
constant 1/2), where H is the regularity index of the underlying instantaneous
volatility process.
