The small and large D limit of Parikh-Wilczek tunneling model for Hawking radiation

In this note, we study both the small and large dimension $D$ limit of the tunneling model of Hwaking radiation proposed by Parikh and Wilczek\cite{Parikh:1999mf}. We confirm that the result $\Gamma \sim e^{\Delta S}$isstillvalidforarbitrary$D>3$\mathrm{.}Thesensiblelarge$D$ limit is given by $\sqrt{D}\ll r_0$ in order to have nonzero radiation. On the other hand, the sensible small $D$ limit is given by taking $D=3+ϵ$ as a continuous parameter. We also explicitly show the leading order correction to the thermal radiation and discuss its connection to the two-dimensional dilaton gravity.