任意深度的复制对称破缺
Replica symmetry breaking at arbitrary depth
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中文总结 AI 辅助
该研究在正温度零外场下的混合偶p自旋伊辛自旋玻璃中，构造出可产生任意深度复制对称破缺（RSB）转变的有限多项式混合，证明了对应扰动的影响集中在接近1的重叠处。
中文摘要 AI 辅助
我们研究正温度、零外场下混合偶p自旋伊辛自旋玻璃中的有限步复制对称破缺（RSB）。对每个整数k≥1，我们构造有限多项式混合，其Parisi测度恰好为k-RSB，且每个此类相在系数的开集上持续存在。更一般地，对任意具有非退化r-RSB Parisi测度的可容许有限多项式协方差Ξ，添加足够大的偶p对应的λq^p，当λ在临界值附近变化时，会产生唯一的r-RSB到(r+1)-RSB的转变。由于此类协方差存在于每个有限RSB深度，这就产生了任意深度的转变。证明的核心思路是：当p很大时，对于远离1的重叠，扰动λq^p可忽略不计，而其影响集中在非常接近1的重叠处。
英文摘要
We study finite-step replica symmetry breaking (RSB) in mixed even $p$-spin Ising spin glasses at positive temperature and zero external field. For every integer $k\geq1$, we construct finite polynomial mixtures whose Parisi measures are exactly $k$-RSB, with each such phase persisting on an open set of coefficients. More generally, for any admissible finite polynomial covariance $Ξ$ with a nondegenerate $r$-RSB Parisi measure, adding $λq^p$ for sufficiently large even $p$ produces, as $λ$ varies near a critical value, a unique $r$-RSB-to-$(r+1)$-RSB transition. Since such covariances exist at every finite RSB depth, this yields transitions of arbitrary depth. The main idea of the proof is that for large $p$, the perturbation $λq^p$ is negligible for overlaps bounded away from $1$, while its effect is concentrated at overlaps very close to $1$.
