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Discrete Math."],"published-print":{"date-parts":[[2023,9,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>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 [Formula: see text]. We establish the existence of fully polynomial-time approximation schemes and samplers with high probability over the random fields if the external fields are independent and identically distributed Gaussians with variance larger than a constant depending only on the inverse temperature and [Formula: see text]. Our methods work more generally for external fields that are large in absolute value with high probability. The main challenge is that such a hypothesis does not rule out the existence of a positive density of vertices at which the external field is small. These regions, which may have connected components of size [Formula: see text], are a barrier to algorithms based on establishing zero-free regions of partition functions 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.<\/jats:p>","DOI":"10.1137\/21m1467389","type":"journal-article","created":{"date-parts":[[2023,7,27]],"date-time":"2023-07-27T05:50:17Z","timestamp":1690437017000},"page":"1610-1629","source":"Crossref","is-referenced-by-count":1,"title":["Approximation Algorithms for the Random Field Ising Model"],"prefix":"10.1137","volume":"37","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4442-2961","authenticated-orcid":true,"given":"Tyler","family":"Helmuth","sequence":"first","affiliation":[{"name":"Mathematical Sciences Department, Durham University, DH1 3LE Durham, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3534-8739","authenticated-orcid":true,"given":"Holden","family":"Lee","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD 21211 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Will","family":"Perkins","sequence":"additional","affiliation":[{"name":"School of Computer Science, Georgia Institute of Technology, Atlanta, GA 30305 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mohan","family":"Ravichandran","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Bogazici University, 34342 Bebek, Istanbul, Turkey."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qiang","family":"Wu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,7,27]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02096933"},{"key":"ref2","doi-asserted-by":"crossref","unstructured":"N. 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