论文标题
离散的集成系统和随机LAX矩阵
Discrete integrable systems and random Lax matrices
论文作者
论文摘要
我们通过考虑其LAX表示，研究了具有随机初始数据的哈密顿综合系统的特性。具体而言，当系统的自由度$ n $时，我们研究相应的宽松矩阵的光谱行为，并根据正确选择的吉布斯度量对初始数据进行采样。我们对指数TODA晶格的状态的极限密度和Volterra晶格进行了确切的描述。为了概括到短距离相互作用（称为INB添加剂和乘法晶格）的概括，将聚焦的Ablowitz-ladik晶格和聚焦的Schur流量进行数值来得出数值状态的密度。对于所有这些系统，我们明确地获得了基态状态的密度。
We study properties of Hamiltonian integrable systems with random initial data by considering their Lax representation. Specifically, we investigate the spectral behaviour of the corresponding Lax matrices when the number $N$ of degrees of freedom of the system goes to infinity and the initial data is sampled according to a properly chosen Gibbs measure. We give an exact description of the limit density of states for the exponential Toda lattice and the Volterra lattice in terms of the Laguerre and antisymmetric Gaussian $β$-ensemble in the high temperature regime. For generalizations of the Volterra lattice to short range interactions, called INB additive and multiplicative lattices, the focusing Ablowitz--Ladik lattice and the focusing Schur flow, we derive numerically the density of states. For all these systems, we obtain explicitly the density of states in the ground states.
