2 The relationship between conductivity and diffusion length is very useful to better understand the transport behavior. By definition, L ( E, t ) = √ D ( E, t ) t , then we can easily find that L ∝ t 1$_{−}$ λ 2 = t $^{α}$, which is verified for all the concentrations simulated, as shown in Fig. 2(a) with a Log-Log plot of diffusion length ( L ) versus time ( t ). Substituting t with L , σ$_{xx}$ scales with L as σ$_{xx}$ ∝ L 2 $^{α$_{−}$}$1 α . By dimensional analysis, σ$_{xx}$ can be expressed as σ$_{xx}$ ( L ) = σ$_{0}$ ( e $^{2}$/h )( L/l ) 2 $^{α$_{−}$}$1 α , (2) Figure 1: (Color online) Diffusion coefficient ( D ) as a function of time ( t ) for different vacancy concentration ( x ) at E $_{F}$. Figure 2: (Color online) (a) Diffusion length ( L ) as a function of time ( t ) for different vacancy concentration ( x ) at E $_{F}$. (b) Scaling exponent α as a function of vacancy concentration ( x ) at E $_{F}$. where σ$_{0}$ is the conductivity in the diffusive regime in unit of e $^{2}$/ ℏ and l is the electron mean free path. It is im- portant to note that such power-law dependence of con- ductivity on diffusion length is different from the usual logarithmic dependence, σ ( L ) = σ$_{0}$ − ( e $^{2}$/ ℏ π $^{2}$) ln ( L/l ) , as predicted by 2D scaling theory of localization [14]. We at- tribute this difference to the distinct nature of the quasi - localization induced by vacancies. The wave-function amplitude of the localized state induced by vacancies de- cays with the distance as 1 /r [6, 8], which is not normal- izable in 2D. Therefore, such quasi -localized state [6, 8] is different from the usual localized state, whose wave function decays exponentially with the distance [14]. It has been predicted that the vacancy-induced quasi - localization will lead to MIT in graphene [6, 8]. However, the critical vacancy concentration where the MIT occurs remains unknown. Here, we will determine the critical vacancy concentration for the MIT. From Fig. 2(a), we derive the scaling exponent, α , from the linear fitting of L versus t , and the resulting α are plotted as a function of x in Fig. 2(b). The curve of α ( x ) can be fit nicely by L ∝ t 1$_{−}$ λ 2 = t $^{α}$, in the whole range of concentration. No- tice that when α = 0 . 5 , λ = 0 , then d D /dt = 0 because D ∝ t − $^{λ}$, which means α = 0 . 5 defines the position of dif- fusive regime. α > 0 . 5 ( λ < 0 ) means the ballistic regime and α < 0 . 5 ( λ > 0 ) means the localization regime. Thus, the power-law scaling holds true for all three transport regimes. α initially decreases rapidly with the increas- ing x , indicating the system quickly leaves the ballistic regime entering the localization regime. The transition point at α = 0 . 5 defines the critical vacancy concentra- tion for MIT, which is found to be x$_{c}$ ≈ 0 . 053% in Fig. 2(b). This value seems to agree quite well with the re- cent experimental result of critical concentration of MIT ( 0 . 08% ) in H-dosed graphene [13]. It implies that similar transport behavior exists in these two disordered systems that both vacancy defect and H impurity create similar localization phenomena in graphene. In addition, the ex- periment suggested other unknown defects rather than H may induce MIT at even lower defect concentrations [13], which might be attributed, at least partly, to vacancies according to our calculation. Next, we examine another criticality of the vacancy- induced MIT in graphene, the mobility edge ( E$_{c}$ ), which plateau often seen in other systems. (2) Beyond D$_{max}$ , the system enters the localization regime, where D de- cays in a power law D ∝ t − $^{λ}$, similar to the behavior of disordered carbon nanotude [19]. This indicates that the system stays in the localization regime ( d D /dt < 0 ) when x ≥ 0 . 1% at large time scale, in agreement with the previous analytical results [6, 8, 10]. (3) D decreases with the increasing x (inset of Fig. 1). This is because higher vacancy concentration leads to stronger scatter- ing, reducing the diffusion coefficient. We note that at a given vacancy concentration, ρ is a constant independent of time. Therefore, the conductivity σ$_{xx}$ ∼ D ρ will have the same power-law scaling with time, σ$_{xx}$ ∝ t − $^{λ}$, as D . Moreover, higher concentration will cause faster decay (larger λ ) in both diffusion coefficient and conductivity.