Physics in nonfixed spatial dimensions via random networks.

Area of Science:

• Condensed Matter Physics
• Quantum Mechanics
• Network Science

Background:

• Random networks lack fixed spatial dimensions, posing challenges for understanding their electronic properties.
• Quantum statistical electronic properties in such systems are not well-understood, particularly regarding dimensionality effects.

Purpose of the Study:

• To investigate the quantum statistical electronic properties of random networks.
• To characterize phenomena like persistent localized states (PLS) and their relation to dimensionality.
• To explore the transition of network properties with increasing connectivity.

Main Methods:

• Analysis of the density of states (DOS) and inverse participation ratio (IPR).
• Calculation of the spatial dimension (D) and its fluctuations.
• Comparison of network properties to graphene lattices and random matrix theory.

Main Results:

• Observed graphene-like density of states and linear dispersion at low connectivity (small R).
• Identified persistent localized states (PLS) at various energies, indicating zero-dimensional (0D) and one-dimensional (1D) quantum orders.
• Found that increasing connectivity (R) leads to a Wigner semicircle DOS and disappearance of PLS.
• Calculated non-integer spatial dimensions (D) with logarithmic dependence on R.

Conclusions:

• Universal physical phenomena can emerge in systems regardless of their spatial dimension.
• The study provides insights into emergent spacetime and quantum gravity concepts.
• Persistent localized states in random networks exhibit dimensional order (0D and 1D).