标题： 扩展在有限连通性模型中产生自旋玻璃序：来自LDPC码理论的严格且直观的方法 显示英文标题

标题： Expansion creates spin-glass order in finite-connectivity models: a rigorous and intuitive approach from the theory of LDPC codes

摘要： 具有许多由大势垒分隔的局部最小值的复杂自由能景观被认为是在各种物理系统中导致玻璃行为的原因。 这是与自旋玻璃中的副本对称性破缺（RSB）相关的一种启发式图景，但RSB仅在某些具有全连接性的平均场模型中被严格验证过。 在本工作中，我们为有限连通性、非欧几里得扩展图上的局部相互作用模型族提供了有限温度自旋玻璃序的严格证明。 为此，我们完全绕过了RSB形式主义，而是利用了这些模型与某些低密度奇偶校验（LDPC）码之间的数学等价性。 我们使用了码扩展性，这是LDPC码的一个属性，它保证了在基态周围存在广泛的能量势垒。 结合一些额外的温和假设，这使我们能够将低温吉布斯态显式地分解为不相交的组件，每个组件都包含一个与景观局部最小值相关的渐近长寿命状态。 每个组件最多携带总权重的指数级小部分，且几乎所有组件都不包含基态——我们将这些一起定义为自旋玻璃序。 该证明是基本的，并以相同的方式处理各种扩展图拓扑，包括现有方法如空腔法失效的短环图。 我们的结果严格适用于足够大的p值的稀释p自旋玻璃，尽管尚未证明，但我们预计我们的假设也适用于更广泛的码族。 受此启发，我们数值研究了两个简单的模型，在随机正则图和双曲空间的正则镶嵌上进行研究。 我们表明，这两个模型随着温度的变化经历了两次转变，分别对应弱遍历性破缺和自旋玻璃序的出现。 显示英文摘要

摘要： Complex free-energy landscapes with many local minima separated by large barriers are believed to underlie glassy behavior across diverse physical systems. This is the heuristic picture associated with replica symmetry breaking (RSB) in spin glasses, but RSB has only been rigorously verified for certain mean-field models with all-to-all connectivity. In this work, we give a rigorous proof of finite temperature spin glass order for a family of models with local interactions on finite-connectivity, non-Euclidean expander graphs. To this end, we bypass the RSB formalism entirely, and instead exploit the mathematical equivalence of such models to certain low-density parity check (LDPC) codes. We use code expansion, a property of LDPC codes which guarantees extensive energy barriers around ground states. Together with mild additional assumptions, this allows us to construct an explicit decomposition of the low-temperature Gibbs state into disjoint components, each hosting an asymptotically long-lived state associated with a local minimum of the landscape. Each component carries at most an exponentially small fraction of the total weight, and almost all components do not contain ground states -- which we take together to define spin-glass order. The proof is elementary, and treats various expanding graph topologies on the same footing, including those with short loops where existing approaches such as the cavity method fail. Our results apply rigorously to diluted p-spin glasses for sufficiently large p, and while unproven, we also expect our assumptions to hold in a broader family of codes. Motivated by this, we numerically study two simple models, on random regular graphs and a regular tesselation of hyperbolic space. We show that both models undergo two transitions as a function of temperature, corresponding to the onset of weak ergodicity breaking and spin glass order, respectively.