标题： 吉布斯测度与多重线性形式 显示英文标题

标题： Gibbs Measures with Multilinear Forms

摘要： 在本文中，我们研究了一类多线性吉布斯测度，其哈密顿量由广义的$\mathrm{U}$统计量给出，并具有一般的基测度。将渐近自由能表示为函数空间上的优化问题，我们得到了副本对称性的必要且充分条件。利用这一点，我们获得了大量感兴趣统计量的弱极限，这包括“局部场/磁化强度”、哈密顿量、全局磁化强度等。一个有趣的结论是在副本对称下对比的普遍弱定律，即$n^{-1}\sum_{i=1}^n c_i X_i\to 0$弱收敛，如果$\sum_{i=1}^n c_i=o(n)$。我们的结果为从极限自由能中出现的优化器提供了概率解释。我们还证明了温度参数下的尖锐相变点的存在，从而推广了仅已知于二次哈密顿量的结果。作为我们证明技术的副产品，我们获得了局部和全局磁化强度的指数集中界，这些结果具有独立兴趣。 显示英文摘要

摘要： In this paper, we study a class of multilinear Gibbs measures with Hamiltonian given by a generalized $\mathrm{U}$-statistic and with a general base measure. Expressing the asymptotic free energy as an optimization problem over a space of functions, we obtain necessary and sufficient conditions for replica-symmetry. Utilizing this, we obtain weak limits for a large class of statistics of interest, which includes the ''local fields/magnetization'', the Hamiltonian, the global magnetization, etc. An interesting consequence is a universal weak law for contrasts under replica symmetry, namely, $n^{-1}\sum_{i=1}^n c_i X_i\to 0$ weakly, if $\sum_{i=1}^n c_i=o(n)$. Our results yield a probabilistic interpretation for the optimizers arising out of the limiting free energy. We also prove the existence of a sharp phase transition point in terms of the temperature parameter, thereby generalizing existing results that were only known for quadratic Hamiltonians. As a by-product of our proof technique, we obtain exponential concentration bounds on local and global magnetizations, which are of independent interest.