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Approximation algorithms for the random-field Ising model
- Publication Year :
- 2021
-
Abstract
- Approximating the partition function of the ferromagnetic Ising model with general external fields is known to be #BIS-hard in the worst case, even for bounded-degree graphs, and it is widely believed that no polynomial-time approximation scheme exists. This motivates an average-case question: are there classes of instances for which polynomial-time approximation schemes exist? We investigate this question for the random field Ising model on graphs with maximum degree $\Delta$. We establish the existence of fully polynomial-time approximation schemes and samplers with high probability over the random fields if the external fields are IID Gaussians with variance larger than a constant depending only on the inverse temperature and $\Delta$. The main challenge comes from the positive density of vertices at which the external field is small. These regions, which may have connected components of size $\Theta(\log n)$, are a barrier to algorithms based on establishing a zero-free region, and cause worst-case analyses of Glauber dynamics to fail. The analysis of our algorithm is based on percolation on a self-avoiding walk tree.<br />Comment: 20 pages
- Subjects :
- Computer Science - Data Structures and Algorithms
Mathematics - Probability
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2108.11889
- Document Type :
- Working Paper