Hypercontractivity implies that for a polynomial $P$ of total degree $d$ in Gaussian variables and $q \geq 2$, we have $$ \|P\|_{L^q} \leq (q-1)^{d/2} \|P\|_{L^2} .$$ Applying Markov inequality for the optimal $q$ yields then for $t \geq C_d$ $$ \mathrm{Prob} \left(|P- \mathbf{E} P| \geq t \sqrt{\mathrm{Var}(P)} \right) \leq \exp(-c_d t^{2/d} ) $$ for some constants $C_d,c_d$.

Hypercontractivity implies that for a polynomial $P$ of total degree $d$ in Gaussian variables and $q \geq 2$, we have $$ \|P\|_{L^q} \leq (q-1)^{d/2} \|P\|_{L^2} .$$ Applying Markov inequality for the optimal $q$ yields then for $t \geq C_d$ $$ \mathrm{Prob} \left(|P- \mathbf{E} P| \geq t \sqrt{\mathrm{Var}(P)} \right) \leq \exp(-c_d t^{2/d} ) $$ for some constants $C_d,c_d$.

Reference: Corollary 5.49 in Aubrun-Szarek, Alice and Bob meet Banach