论文标题
在蜘蛛网和风扇网络上跨越树木的普遍性和确切的有限尺寸校正
Universality and exact finite-size corrections for spanning trees on cobweb and fan networks
论文作者
论文摘要
普遍性是关键现象理论的基石。在大多数系统中，尤其是在热力学极限中，它在大多数系统中得到了充分的了解。有限大小的系统提出了其他挑战。即使在低维度中，边缘和角度对自由能和响应功能的贡献也不太理解。问题是如何在同一普遍性类别的系统中校正到缩放中如何维持普遍性，但具有截然不同的角度几何形状。 2D几何形状提供了最简单的示例，这些示例可以在有或没有角落构造。为了调查角落的存在和不存在如何表现普遍性，我们分析了两个有限系统上的生成函数，即蜘蛛网和风扇网络。鉴于有限尺寸的蜘蛛网没有弯道，而球迷有四个。为了回答，我们诉诸于Ivashkevich-izmailian-Hu方法，该方法将不同网络的生成函数从具有扭曲的边界条件的单个分区函数方面统一。这种统一的方法表明，对风扇网络总和的各个角落自由能的贡献至零，因此它与网络的贡献完全匹配。在每种情况下，通过替代方式为两个网络建立的结果验证算法的声音。它的实用性范围通过其应用于迄今未解决问题的应用，即生成函数对数的确切渐近扩展以及FAN和CobWeb几何形状的共形函数。因此，通用性难题的解决表明了算法的力量，并在将来打开了新的应用程序。
Universality is a cornerstone of theories of critical phenomena. It is well understood in most systems especially in the thermodynamic limit. Finite-size systems present additional challenges. Even in low dimensions, universality of the edge and corner contributions to free energies and response functions is less well understood. The question arises of how universality is maintained in correction-to-scaling in systems of the same universality class but with very different corner geometries. 2D geometries deliver the simplest such examples that can be constructed with and without corners. To investigate how the presence and absence of corners manifest universality, we analyze the spanning tree generating function on two finite systems, namely the cobweb and fan networks. We address how universality can be delivered given that the finite-size cobweb has no corners while the fan has four. To answer, we appeal to the Ivashkevich-Izmailian-Hu approach which unifies the generating functions of distinct networks in terms of a single partition function with twisted boundary conditions. This unified approach shows that the contributions to the individual corner free energies of the fan network sum to zero so that it precisely matches that of the web. Correspondence in each case with results established by alternative means for both networks verifies the soundness of the algorithm. Its range of usefulness is demonstrated by its application to hitherto unsolved problems-namely the exact asymptotic expansions of the logarithms of the generating functions and the conformal partition functions for fan and cobweb geometries. Thus, the resolution of a universality puzzle demonstrates the power of the algorithm and opens up new applications in the future.
