Directed Abelian algebras and their application to stochastic models

Area of Science:

• Statistical Mechanics
• Algebraic Methods
• Complex Systems

Background:

• Directed acyclic graphs (DAAs) can be associated with Abelian algebras.
• These algebras are semisimple and depend on parameters.
• DAAs can define Hamiltonians for stochastic processes.

Purpose of the Study:

• To apply Directed Abelian Algebras (DAAs) to sandpile models.
• To analyze the spectral properties and dynamics of these systems.
• To investigate avalanche behavior in 1D and 2D lattices.

Main Methods:

• Algebraic construction of DAAs and associated Hamiltonians.
• Calculation of spectra and ground-state wave functions.
• Finite-size scaling analysis for D-dimensional lattices.
• Monte Carlo simulations for 2D sandpile models.

Main Results:

• DAA-based Hamiltonians exhibit gapless spectra with critical dynamic exponent z=D for D-dimensional lattices.
• In 1D, particle-conserving DAAs lead to avalanches in the random walker universality class (sigma = 3/2).
• 1D avalanches show particle depletion at the source and enrichment at the end.
• 2D simulations yield sigma = 1.780 ± 0.005.

Conclusions:

• DAAs provide a powerful algebraic framework for studying stochastic processes and complex systems like sandpiles.
• The study establishes a connection between algebraic structures and critical phenomena in physical models.
• The findings offer insights into the universality classes and scaling behaviors of sandpile avalanches.