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<p>In quenched disordered systems, the existence of ordering is generally believed to be only possible in the weak disorder regime (disregarding models of spin-glass type). In particular, sufficiently large random fields is expected to prohibit any finite temperature ordering. Here, we prove that this is not necessarily true, and show rigorously that for physically relevant systems in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {Z}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(d\ge 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, disorder can induce ordering that is <i>infinitely stable</i>, in the sense that (1) there exists ordering at arbitrarily large disorder strength and (2) the transition temperature is asymptotically nonzero in the limit of infinite disorder. Analogous results can hold in 2 dimensions provided that the underlying graph is non-planar (e.g., <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {Z}^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> sites with nearest and next-nearest neighbor interactions).</p>

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Infinite Stability in Disordered Systems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月26日

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  • Andrew C. Yuan,
  • Nick Crawford

摘要

In quenched disordered systems, the existence of ordering is generally believed to be only possible in the weak disorder regime (disregarding models of spin-glass type). In particular, sufficiently large random fields is expected to prohibit any finite temperature ordering. Here, we prove that this is not necessarily true, and show rigorously that for physically relevant systems in \(\mathbb {Z}^d\) Z d with \(d\ge 3\) d ≥ 3 , disorder can induce ordering that is infinitely stable, in the sense that (1) there exists ordering at arbitrarily large disorder strength and (2) the transition temperature is asymptotically nonzero in the limit of infinite disorder. Analogous results can hold in 2 dimensions provided that the underlying graph is non-planar (e.g., \(\mathbb {Z}^2\) Z 2 sites with nearest and next-nearest neighbor interactions).