Delocalization-localization dynamical phase transition of random walks on graphs

Area of Science:

• Statistical physics
• Graph theory
• Dynamical systems

Background:

• Random walks are fundamental processes on graphs.
• Understanding large deviations is crucial for complex systems.
• Dynamical phase transitions (DPTs) offer insights into system behavior.

Purpose of the Study:

• To investigate the large deviations of a local observable in random walks on graphs.
• To identify and characterize dynamical phase transitions in these systems.
• To analyze the impact of graph topology on these transitions.

Main Methods:

• Analysis of random walks on two models of connected, undirected graphs.
• Application of large deviation theory in the thermodynamic limit.
• Analytical characterization of finite-size crossover scaling functions.

Main Results:

• A first-order dynamical phase transition (DPT) was proven for the observable.
• The DPT is interpreted as a coexistence of localized and delocalized paths.
• The scaling function for the localized-delocalized crossover was analytically derived.
• The DPT was shown to be robust against changes in graph topology.

Conclusions:

• First-order DPTs are a key feature of random walks on certain graphs.
• Graph topology primarily influences the crossover regime, not the transition itself.
• These findings suggest DPTs may occur in random walks on infinite random graphs.