Anomalous Geodesics in the Inhomogeneous Corner Growth Model

Area of Science:

• Probability theory
• Statistical mechanics
• Stochastic processes

Background:

• Last-passage percolation models are fundamental in probability and statistical physics.
• Understanding geodesics and interface dynamics is crucial for analyzing complex systems.
• Inhomogeneity introduces novel behaviors not seen in uniform models.

Purpose of the Study:

• To investigate Busemann functions, semi-infinite geodesics, and competition interfaces in exactly solvable last-passage percolation with inhomogeneous exponential weights.
• To identify and characterize new phenomena arising from inhomogeneity in these models.
• To analyze the behavior of second-class customers in inhomogeneous continuous-time queues.

Main Methods:

• Exact solution of last-passage percolation models.
• Analysis of Busemann functions and geodesic properties.
• Modeling of competition interfaces and queueing systems.

Main Results:

• Discovery of novel Busemann functions linked to flat regions and thin rectangles.
• Observation of semi-infinite geodesics with intervals of asymptotic directions.
• Identification of non-trivial axis-directed geodesics and intervals lacking geodesic directions.
• Characterization of a new dichotomy for competition interfaces and second-class customers in inhomogeneous queues, where customers either get trapped or move at positive speed.

Conclusions:

• Inhomogeneity in last-passage percolation leads to complex and novel geodesic behaviors.
• The study reveals new dynamics in competition interfaces and customer flow within inhomogeneous queueing systems.
• These findings advance the understanding of stochastic processes in non-uniform environments.