Linking physics and algorithms in the random-field Ising model

TL;DR

This paper investigates the scaling properties and dynamics of the push-relabel algorithm in solving the random-field Ising model, revealing connections between algorithmic behavior and the physical properties of the system.

Contribution

It provides a detailed analysis of the algorithm's dynamics, critical behavior near phase transitions, and links between auxiliary field evolution and RFIM physics.

Findings

Algorithm dynamics relate to two-species annihilation at small fields

Correlation length diverges at zero disorder in 1D and 2D

Time to solution diverges near critical point with specific critical exponents

Abstract

The energy landscape for the random-field Ising model (RFIM) is complex, yet algorithms such as the push-relabel algorithm exist for computing the exact ground state of an RFIM sample in time polynomial in the sample volume. Simulations were carried out to investigate the scaling properties of the push-relabel algorithm. The time evolution of the algorithm was studied along with the statistics of an auxiliary potential field. At very small random fields, the algorithm dynamics are closely related to the dynamics of two-species annihilation, consistent with fractal statistics for the distribution of minima in the potential (``height''). For $d=1,2$, a correlation length diverging at zero disorder sets a cutoff scale for the magnitude of the height field; our results are most consistent with a power-law correction to the exponential scaling of the correlation length with disorder in $d=2$. Near the ferromagnetic-paramagnetic transition in $d=3$, the time to find a solution diverges with a dynamic critical exponent of $z=0.93\pm0.06$ for a priority queue version and $z=0.43\pm0.06$ for a first-in first-out queue version of the algorithm. The links between the evolution of auxiliary fields in algorithmic time and the static physical properties of the RFIM ground state provide insight into the physics of the RFIM and a better understanding of how the algorithm functions.