均匀磁场中的分支随机游走：磁化强度集中与重叠分布
The branching random walk in a uniform magnetic field : magnetization concentration and overlap distributions
- Sorbonne Université, Sorbonne Paris Cité, CNRS, Laboratoire de Probabilités Statistique et Modélisation, LPSM（索邦大学、索邦巴黎城、法国国家科学研究中心、概率统计与建模实验室）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究均匀磁场中的高斯二元分支随机游走（BRW），利用一般BRW的已有结果分析其外磁场下的性质，证明低温磁化强度的强集中性，并通过双复制大偏差论证得到Ising超立方体型重叠的分布。
AI中文摘要：
向平均场自旋玻璃模型中添加均匀外磁场通常需要针对该模型进行全新分析。我们所考虑的无序系统对应高斯二元分支随机游走（BRW），其研究思路源自Derrida和Spohn的工作[21]，Jagannath已从统计物理学视角对其开展研究[22]，而我们证明情况并非如此：一个基本观察结果——所得哈密顿量仍是BRW，只是位移独立但非同分布——使我们能够利用现有一般BRW的研究成果。结合一般BRW的经典与最新成果（Biggins[3]、Chauvin和Rouault[13]、Mallein[25]），可对外磁场下该模型获得基本完整的认识：基态、自由能、一步复制对称破缺（1-RSB）相变，以及Gibbs权重服从Poisson-Dirichlet统计的系谱重叠极限分布。随后，我们证明低温下Gibbs测度下磁化强度的强集中结果，给出其显式最优值。结果表明，单复制陈述不足以控制两个独立采样构型间的经典Ising重叠：一个基本反例（注4.1）显示，每个复制磁化强度的集中本身并不决定它们的联合相关性。我们通过开发双复制大偏差论证解决此问题——该论证基于经典类型方法和Shannon熵的次可加性，采取针对一对构型的联合经验类型的一致Chernoff界形式，与双复制浓度估计匹配——我们用其得到第二类（Ising）超立方体型重叠的分布。
英文摘要：
Adding a uniform external magnetic field to a mean-field spin-glass model usually requires a new analysis specific to the model. The disordered system we consider corresponds to the Gaussian binary branching random walk (BRW) - in the spirit of Derrida and Spohn [21] and studied from the statistical-physics point of view by Jagannath [22] - and we prove that this is not the case : a single elementary observation - that the resulting Hamiltonian is still a BRW, now with independent, but non-identically distributed, displacements - allows us to use the available results for general BRW. Combining classical and recent results on general BRW (Biggins [3], Chauvin and Rouault [13], Mallein [25]), one obtains an essentially complete picture of the model in an external magnetic field : the ground state, the free energy, the one-step replica symmetry breaking (1-RSB) transition, and the limiting genealogical overlap distribution with Poisson-Dirichlet statistics for the Gibbs weights. We then prove a strong concentration result for the magnetization under the Gibbs measure at low temperature, giving its explicit optimal value. It turns out that this one-replica statement is not enough to control the classical Ising overlap between two independently sampled configurations : an elementary counterexample (Remark 4.1) shows that concentration of each replica's magnetization does not, by itself, determine their joint correlation. We address this by developing a two-replica large-deviation argument - resting on the classical method of types and the subadditivity of Shannon entropy, and taking the form of a uniform Chernoff bound over the joint empirical type of a pair of configurations, matched against a two-replica concentration estimate - which we use to obtain the distribution of a second (Ising) hypercube-type overlap.
