Fractal-driven universality in quantum transport through mesoscopic devices

We examine the impact of spatial fractality on universal quantum transport in mesoscopic systems featuring Sierpiński carpet scattering geometries. Transport calculations are performed using the nonequilibrium Green's function formalism within the Landauer-Büttiker framework, allowing for a detailed statistical characterization of conductance and shot noise power fluctuations. Our analysis reveals that the fractal dimension of these fluctuations increases systematically with the iteration depth of the structure, reflecting enhanced complexity in quantum interference patterns. A direct and robust correlation is established between the Hausdorff dimension of the scattering region and the effective fractal dimension of transport observables, independent of the Wigner-Dyson universal symmetry class (GOE, GUE, and GSE). This provides compelling evidence that scattering across internal lacunae constitutes the dominant mechanism shaping transport behavior in the universal regime. Moreover, we find that the density of local extrema increases while the correlation length decreases with increasing fractality; however, the number of extrema per correlation length remains remarkably with negligible variation. Notably, the autocorrelation functions progressively deviate from the standard Lorentzian form, indicating that geometric self-similarity introduces nontrivial modifications to the spectral correlations predicted by Random Matrix Theory.