The Thermodynamic Limit in Mean Field Spin Glass Models
The Thermodynamic Limit in Mean Field Spin Glass Models
复制标题
平均场自旋玻璃模型的热力学极限
DOI：
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
F. Toninelli
中科院分区：
文献类型：
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作者：
F. Guerra;F. Toninelli
摘要： 对于一大类平均场无序模型，例如谢林顿-柯克帕特里克模型和德里达p自旋模型，我们提出了一个简单的策略来证明热力学无限体积极限的存在和唯一性。主要的论点是基于一个由N个自旋位组成的大系统和两个相似但独立的子系统之间的平滑内插，这两个子系统分别由N1和N2位组成，其中N1+N2=N。自由能的猝灭平均值对于系统的大小来说是次加的。这立即使每个位置的自由能在无限大的体积极限内收敛。此外，一个简单的论证，基于测量的集中度，给出了关于外部噪声的几乎必然的收敛。类似的结果也适用于每个位置的基态能量。
Abstract: We present a simple strategy in order to show the existence and uniqueness of the infinite volume limit of thermodynamic quantities, for a large class of mean field disordered models, as for example the Sherrington-Kirkpatrick model, and the Derrida p-spin model. The main argument is based on a smooth interpolation between a large system, made of N spin sites, and two similar but independent subsystems, made of N1 and N2 sites, respectively, with N1+N2=N. The quenched average of the free energy turns out to be subadditive with respect to the size of the system. This gives immediately convergence of the free energy per site, in the infinite volume limit. Moreover, a simple argument, based on concentration of measure, gives the almost sure convergence, with respect to the external noise. Similar results hold also for the ground state energy per site.