As an unusual type of anomalous diffusion behavior, namely (transient) superballistic transport, has been experimentally observed recently, but it is not yet well understood. In this paper, we investigate the white noise effect (in the Markov approximation) on quantum diffusion in one-dimensional tight-binding models with a periodic, disordered, and quasiperiodic region of size L attached to two perfect lattices at both ends in which the wave packet is initially located at the center of the sublattice. We find that in a completely localized system, inducing noise could delocalize the system to a desirable diffusion phase. This controllable system may be used to investigate the interplay of disorder and white noise, as well as to explore an exotic quantum phase.

We construct a new class of [Formula: see text]-dimensional black hole solutions with Toroidal horizons in [Formula: see text] gravity framework which is surrounded by quintessence and string cloud configuration, whose asymptotic structures are determined by the quintessential state parameter [Formula: see text] and dimension’s parameter [Formula: see text]. We point out that the asymptotic behavior of the obtained solutions is neither asymptotically flat nor (A)dS when we have [Formula: see text] and [Formula: see text]. Regarding the cosmological constant as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume, we investigate the critical behavior of our black hole solutions. In [Formula: see text] and [Formula: see text], the P–V diagrams are more complex than that of the standard Van der Waals. These diagrams behave like the Born–Infeld-AdS P–V diagrams. When [Formula: see text] and [Formula: see text], the P–V diagrams behave like the standard Van der Waals system (In this work [S. Gunasekaran, D. Kubiznaak and R. B. Mann, Extended phase space thermodynamics for charged and rotating black holes and Born–Infeld vacuum polarization, J. High Energy Phys. (2012).] [S. Gunasekaran, D. Kubiznaak and R.B. Mann, Extended phase space thermodynamics for charged and rotating black holes and Born–Infeld vacuum polarization, J. High Energy Phys. (2012)], the authors believe that the BTZ black holes, contrary to the higher-dimensional case, do not show any critical behavior. In this work, we have showed that the three-dimensional (3D) black holes can also have critical behavior and coincide with those of the Van der Waals system). Besides, we have analyzed the concept of effective potential barrier by transforming the radial equation of motion into standard Schrodinger form. We figure out the effect of the coupling constant [Formula: see text], the string parameter b, and the quintessential state parameter [Formula: see text] on the height of the potential barrier and the other thermodynamical quantities like temperature and heat capacity. Then, we study the quasinormal modes (QNMs) of 3D black holes in the [Formula: see text] context. For this purpose, we use the WKB approximation method upto third-order corrections. We have shown the perturbation’s decay in corresponding diagrams when the string parameter b changes.