AI 中文总结
本研究通过蒙特卡洛模拟和场论，证明一维长程随机环图上的Sparse-SAW与干净超扩散Lévy-SAW属同一普适类，确立随机图为提取该类临界指数的有效平台。
AI 中文摘要
具有长程跳跃的随机游走可驱动超扩散输运，用有效长程动力学算子替代普通扩散，这类超扩散动力学也是临界现象的核心，特别是具有长程跳跃统计的自回避游走（self-avoiding walk, SAW），即Lévy-SAW。本研究探讨当长程连接本身变为随机时，临界行为会受到怎样的影响。我们研究一维长程随机环图上的自回避游走，其中键通过伯努利概率~|i-j|^{-(1+σ)}独立生成，将这类游走命名为Sparse-SAW。相同的随机键同时负责长程超扩散输运和淬火无序，且两者均由单一参数σ控制，该问题超出了传统的Harris和Weinrib-Halperin框架。通过大规模蒙特卡洛模拟和高斯截断场论，我们证明Sparse-SAW属于与干净超扩散Lévy-SAW相同的普适类。随机键产生短程无关联和长程关联的质量无序，同时生成长程动力学算子；在粗粒化过程中，后者占主导地位，恢复了干净临界行为。我们的研究表明，完全非高斯伯努利统计可能会产生超出传统淬火无序理论的无序物理，同时确立随机图作为提取干净超扩散Lévy-SAW普适类临界指数的有效平台。
英文摘要
Random walks with long-range jumps can drive superdiffusive transport, replacing ordinary diffusion with an effective long-range kinetic operator. Such superdiffusive kinetics is also central to critical phenomena, notably the self-avoiding walk with long-range jump statistics, or Lévy-SAW. This work investigates how the critical behavior is affected when the long-range connectivity itself becomes random. We study self-avoiding walks (SAWs) on a one-dimensional long-range random ring graph, where bonds are independently generated with Bernoulli probability $\sim|i-j|^{-(1+σ)}$. We term this walk Sparse-SAW. The same random bonds are responsible for both long-range superdiffusive transport and quenched disorder, with both simultaneously controlled by the single parameter $σ$, placing the problem beyond the conventional Harris and Weinrib-Halperin frameworks. Through large-scale Monte Carlo simulations and a Gaussian-truncated field theory, we show that Sparse-SAW belongs to the same universality class as the clean superdiffusive Lévy-SAW. The random bonds generate short-range uncorrelated and long-range correlated mass disorder while simultaneously producing the long-range kinetic operator. Under coarse-graining, the latter dominates, restoring the clean critical behavior. Our study suggests that the full non-Gaussian Bernoulli statistics may lead to disorder physics beyond the conventional theory of quenched disorder, while establishing random graphs as an efficient platform for extracting the critical exponents of the clean superdiffusive Lévy-SAW universality class.
Comments12 (7+5) pages, 2 Figures. Comments are welcome
