Directed abelian algebras and their applications to stochastic models

TL;DR

This paper introduces directed abelian algebras (DAA) linked to directed acyclic graphs, enabling algebraic analysis of stochastic models like sandpiles, revealing critical spectra and avalanche behaviors in various dimensions.

Contribution

It develops the theory of DAA, applies it to stochastic models including sandpiles, and analyzes their spectral properties and avalanche dynamics in different dimensions.

Findings

Spectrum is gapless with dynamic exponent z=D in D-dimensional lattices.

In 1D, avalanches follow the random walker universality class with exponent 3/2.

In 2D, avalanche size distribution exponent is approximately 1.782.

Abstract

To each directed acyclic graph (this includes some D-dimensional lattices) one can associate some abelian algebras that we call directed abelian algebras (DAA). On each site of the graph one attaches a generator of the algebra. These algebras depend on several parameters and are semisimple. Using any DAA one can define a family of Hamiltonians which give the continuous time evolution of a stochastic process. The calculation of the spectra and ground state wavefunctions (stationary states probability distributions) is an easy algebraic exercise. If one considers D-dimensional lattices and choose Hamiltonians linear in the generators, in the finite-size scaling the Hamiltonian spectrum is gapless with a critical dynamic exponent $z = D$. One possible application of the DAA is to sandpile models. In the paper we present this application considering one and two dimensional lattices. In the one dimensional case, when the DAA conserves the number of particles, the avalanches belong to the random walker universality class (critical exponent $\sigma_{\tau} = 3/2$). We study the local densityof particles inside large avalanches showing a depletion of particles at the source of the avalanche and an enrichment at its end. In two dimensions we did extensive Monte-Carlo simulations and found $\sigma_{\tau} = 1.782 \pm 0.005$.