Sparre-Andersen theorem with spatiotemporal correlations.

Area of Science:

• Probability theory
• Statistical physics
• Random walk theory

Background:

• The Sparre-Andersen theorem addresses the universality of first-passage-time distributions in 1D random walks.
• This theorem is crucial for understanding how walkers reach boundaries, irrespective of jump distribution details.
• Existing models often focus on the number of steps, not the time taken, which can be critical in physical scenarios.

Purpose of the Study:

• To extend the Sparre-Andersen theorem to scenarios where step time correlates with step length.
• To develop a modified proof applicable to more general random walk processes.
• To investigate deviations from normal scaling in first-passage-time distributions when time and step variables are linked.

Main Methods:

• Modification of the Sparre-Andersen proof technique.
• Development of a generalized framework for correlated step times and lengths.
• Application to a specific 2D physical process as a case study.

Main Results:

• A generalized Sparre-Andersen theorem is presented for correlated random walks.
• Deviations from normal scaling in first-passage-time distributions are demonstrated in a 2D example.
• The findings highlight the importance of considering time-step correlations in random walk analysis.

Conclusions:

• The study successfully extends the Sparre-Andersen theorem to a broader class of random walks.
• The results emphasize the impact of time-step correlations on first-passage-time distributions.
• This work provides a more realistic model for physical systems where step duration is variable.