Metric networks are network-shaped, one-dimensional structures on which one can solve differential equations to simulate a wide range of physical systems including conjugated molecules, photonic crystals, quantum mechanics in waveguide networks, and acoustic metamaterials. More concretely, a metric network is a network whose edges are each assigned a notion of length and a coordinate describing position. One can then define function spaces and differential operators on these objects to model the aforementioned systems. Recent software advancements have made it feasible to analyze partial differential equations on large, compact metric networks with a vast array of structures. Here, we generate compact metric network structures using the spatial tessellations of two-dimensional hyperuniform point patterns, which have suppressed large-scale density fluctuations relative to typical disordered point patterns. This choice of structure is inspired by the exotic physical properties of network materials with these structures in other contexts. Then, we characterize the eigenvalue spectrum structure of the differential Laplace operator on these networks. In particular, we find that gaps can form in the eigenvalue spectra of these networks whose widths increase when the distribution of edge lengths is narrow and as the number of triangular faces increases. Importantly, many of the structures we consider are realizable in Euclidean space, meaning they are well-suited for practical applications in, e.g., metamaterial design. This work can thus be used to inform the design of metric network-based systems with spectral gaps with tunable widths and locations.
Experimental study of the correlation length of critical-current fluctuations in the presence of surface disorder: Probing vortex long-range interactions
我们报告的临界电流和电压噪声测量铌条在超导状态下，在存在的体积涡旋晶格（$B < B_{C2}$）和在表面超导状态（$B_{c2}< B < B_{C3}$）。对于均匀表面，电流波动的相关长度可以与涡流表面不稳定性的电磁趋肤深度相关联。通过低能量照射对表面状态的改变会引起临界电流和噪声的强烈改变。在频谱域中出现的拐角频率可以与人工噪声的低波矢量相联系。由于后者只发生在$B < B_{C2}$，我们建议，长程相互作用允许的相关长度延伸到由表面形貌施加的值。
Probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension $d=1,2,3$ via a Monte-Carlo procedure in the disorder
Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder
本文研究了具有独立同分布有界变量的随机边长和分支数为$ K \geq 2 $的根度量树图上的拉普拉斯算子。我们的主要结果是弱无序时绝对连续谱的稳定性。在讨论中的一个有用的工具是一个函数，它表示一个方向性的传输幅度到无穷大，并形成一个推广的Weyl-Titchmarsh函数的树。主要结果的证明依赖于弱无序极限中这个量的波动范围的上界。
Probing phase transitions of finite directed polymers near a corrugated wall via two-replica analysis
Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京数学科学与工程研究院）；Laboratory of Complex Networks, Center for Neurophysics and Neuromorphic Technologies（复杂网络实验室，神经物理与类脑技术中心）
