We consider a system of harmonic oscillators with short range interactions and we study their correlation functions when the initial data is sampled with respect to the Gibbs measure. Such correlation functions display rapid oscillations that travel through the chain. We show that the correlation functions always have two fastest peaks which move in opposite directions and decay at rate \(t^{-\frac{1}{3}}\) for position and momentum correlations and as \(t^{-\frac{2}{3}}\) for energy correlations. The shape of these peaks is asymptotically described by the Airy function. Furthermore, the correlation functions have some non generic peaks with lower decay rates. In particular, there are peaks which decay at rate \(t^{-\frac{1}{4}}\) for position and momentum correlators and with rate \(t^{-\frac{1}{2}}\) for energy correlators. The shape of these peaks is described by the Pearcey integral. Crucial for our analysis is an appropriate generalisation of spacings, i.e. differences of the positions of neighbouring particles, that are used as spatial variables in the case of nearest neighbour interactions. Using the theory of circulant matrices we are able to introduce a quantity that retains both localisation and analytic viability. This also allows us to define and analyse some additional quantities used for nearest neighbour chains. Finally, we study numerically the evolution of the correlation functions after adding nonlinear perturbations to our model. Within the time range of our numerical simulations the asymptotic description of the linear case seems to persist for small nonlinear perturbations while stronger nonlinearities change shape and decay rates of the peaks significantly.

我们考虑具有短程相互作用的谐波振荡器系统，并且当针对吉布斯测度采样初始数据时，我们研究它们的相关函数。这样的相关函数显示出在链中传播的快速振荡。我们显示出，对于位置和动量相关性，相关函数始终具有两个最快的峰，它们沿相反的方向移动并以速率\（t ^ {-\ frac {1} {3}} \）衰减，并且为\（t ^ {- \ frac {2} {3}} \）进行能量相关。这些峰的形状由艾里函数渐近描述。此外，相关函数具有一些具有较低衰减率的非通用峰。特别是，有些峰以速率\（t ^ {-\ frac {1} {4}} \）衰减。用于位置和动量相关器，速率为\（t ^ {-\ frac {1} {2}} \）用于能源相关器。这些峰的形状由Pearcey积分描述。对于我们的分析，至关重要的是适当地概括间距，即相邻粒子位置的差异，在最邻近相互作用的情况下用作空间变量。使用循环矩阵的理论，我们能够引入一个既保留局部性又保留分析可行性的量。这也使我们能够定义和分析一些用于最近邻居链的额外数量。最后，在对模型添加非线性扰动后，我们对相关函数的演化进行了数值研究。