Giant leaps and long excursions: Fluctuation mechanisms in systems with long-range memory

Area of Science:

• Statistical Physics
• Complex Systems
• Probability Theory

Background:

• Stochastic processes are fundamental in modeling complex systems.
• Long-range memory introduces dependencies that deviate from standard Markovian assumptions.
• Understanding large deviations is crucial for characterizing system behavior under rare events.

Purpose of the Study:

• To analyze large deviations of time-averaged quantities in stochastic processes with long-range memory.
• To identify and explain mechanisms causing unusual fluctuation behavior due to memory.
• To demonstrate the generic nature of these mechanisms in various memory-dependent models.

Main Methods:

• Analysis of large deviations in stochastic processes.
• Investigation of specific models like the elephant random walk and its Gaussian variant.
• Examination of a non-Markovian simple exclusion process.

Main Results:

• Identified two distinct mechanisms for unusual fluctuation behavior in memory-dependent dynamics.
• Demonstrated that long-range memory can lead to large-deviation principles with reduced speeds.
• Showed that memory effects can result in nonanalytic rate functions.

Conclusions:

• The identified mechanisms for unusual fluctuations are generic in memory-dependent dynamics.
• Stochastic processes with long-range memory exhibit fundamentally different large-deviation properties compared to Markovian systems.
• These findings offer new insights into the statistical behavior of systems with complex temporal dependencies.