[Paper Review] Linking physics and algorithms in the random-field Ising model

Jan H. Meinke, Middleton, A. Alan|arXiv (Cornell University)|Feb 21, 2005

Theoretical and Computational Physics5 references3 citations

TL;DR

This paper links the dynamics of the push-relabel algorithm—used to compute the exact ground state of the random-field Ising model (RFIM)—to physical properties of the RFIM, revealing that algorithmic time evolution mirrors two-species annihilation processes at low disorder. It finds that the algorithm's runtime diverges near the zero-temperature phase transition with dynamic critical exponents z ≈ 0.93 (priority queue) and z ≈ 0.43 (FIFO queue) in d=3, while fractal statistics of the auxiliary potential field reflect the underlying physics of domain formation and critical scaling.

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.

Motivation & Objective

• To understand the connection between algorithmic dynamics of the push-relabel method and the physical ground state structure of the RFIM.
• To investigate how the auxiliary potential field in the push-relabel algorithm reflects the energy landscape and phase transitions in the RFIM.
• To analyze the scaling behavior of the algorithm’s runtime near the zero-temperature paramagnetic-ferromagnetic transition in different dimensions.
• To examine the relationship between the algorithm’s dynamics at low disorder and two-species annihilation processes in statistical physics.
• To determine whether fractal statistics of the potential field correlate with critical behavior and correlation length divergence in the RFIM.

Proposed method

• The push-relabel (PR) algorithm is applied to compute the exact ground state of the RFIM, using auxiliary fields (the 'height' or potential field) to guide spin configurations.
• The dynamics of the auxiliary potential field are studied as a function of algorithmic time, particularly focusing on the evolution of excesses (positive) and sinks (negative) in the field.
• The algorithm is implemented with both FIFO and priority queue update rules to compare their scaling behavior and dynamic critical exponents.
• The statistics of the final potential field are analyzed, including the distribution of height values and their dependence on system size and disorder strength.
• The fractal dimension of the sink distribution is estimated via correlation analysis, especially in the limit of small disorder.
• Scaling analysis is performed on the runtime peak location and height field distribution to test predictions from RFIM critical theory, including correlation length divergence and power-law corrections.

Experimental results

Research questions

• RQ1 How do the dynamics of the auxiliary potential field in the push-relabel algorithm relate to physical processes such as two-species annihilation in low-disorder RFIM?
• RQ2 What is the scaling behavior of the push-relabel algorithm’s runtime near the zero-temperature phase transition in d=1, 2, and 3?
• RQ3 How do the fractal statistics of the final potential field reflect the underlying physics of domain formation in the RFIM ground state?
• RQ4 To what extent does the distribution of the potential field near criticality match theoretical predictions for the RFIM’s correlation length and critical exponents?
• RQ5 How do different queueing strategies (FIFO vs. priority queue) affect the dynamic critical exponent z of the algorithm’s runtime?

Key findings

• At low disorder, the push-relabel algorithm’s dynamics closely resemble two-species annihilation, with mobile positive excesses coalescing and immobile negative excesses forming fractal sinks.
• In d=1, the fractal dimension of the sink distribution is D ≈ 0.50, consistent with random walk annihilation theory.
• In d=2, the fractal dimension is D ≈ 0.40, and in d=3, it is D < 0.2, indicating decreasing spatial complexity with dimension.
• For d=2, the peak in algorithm runtime scales with system size in a way consistent with power-law corrections to the exponential divergence of the correlation length ξ(Δ) ∼ exp(cΔ⁻²).
• In d=3, near the critical disorder Δc, the runtime diverges with dynamic critical exponents z ≈ 0.93 (priority queue) and z ≈ 0.43 (FIFO queue), matching known RFIM critical exponents.
• The potential field distribution P(ui) is nearly constant for ui ≪ ξ in d=3, and P(ui) ∝ ui⁻¹/² for u ≪ ξ in d=1, indicating universal scaling near criticality.