AI 中文总结
本文研究圆柱表面上二维单组分等离子体配分函数的大$N$展开，通过变量变换至环形液滴模型，利用拉普拉斯方法和欧拉-麦克劳林公式处理硬壁情况，发现对数展开中无$\log N$项。
AI 中文摘要
二维单组分等离子体在单体势与电荷数成正比的情况下会形成液滴。我们感兴趣的是研究相应的配分函数的大$N$形式，其中等离子体被限制在圆柱表面上。单体势（在技术限制范围内）允许在圆柱轴方向上任意变化，而耦合被限制在精确可解值$\beta = 2$。我们证明了可以通过变量变换，将问题转化为环形液滴等离子体模型，但需乘以某个雅可比因子。该因子可解释为某个线性统计量的特征函数，这使我们的研究目标得以推广，即同时分析环形液滴等离子体模型特征函数的大$N$形式。虽然在软壁边界条件下这是众所周知的，但我们的分析（基于应用于某些积分的拉普拉斯方法以及欧拉-麦克劳林求和公式）涵盖了液滴边界处或内部存在一个或两个硬壁的情况，并表明软边公式在这些情况下不再成立。利用这一点来研究圆柱配分函数的大$N$展开，需要知道环形情况下的相同展开。在液滴内部存在两个硬壁的情况下，这一展开无法获得，我们转而通过直接分析来处理。与环形配分函数的情况不同，我们发现圆柱配分函数对数的大$N$展开始终没有正比于$\log N$的项，无论是否存在硬壁。
英文摘要
The two-dimensional one-component plasma forms a droplet in the case that the one-body potential is proportional to the number of charges. Our interest is in studying the large $N$ form of the corresponding partition function, with the plasma confined to the surface of the cylinder. The one-body potential is (up to technical restrictions) allowed to be arbitrary in the direction of the axis of the cylinder, whereas the coupling is restricted to the exactly solvable value $β= 2$. It is demonstrated that a change of variables can be made to an annular droplet plasma model, up to a certain Jacobian factor. The latter can be interpreted as the characteristic function of a certain linear statistic, which generalises the original aim of our study to also analysing the large $N$ form of the characteristic function for the annular droplet plasma model. While this is well known in the case of soft wall boundary conditions, our analysis (based on Laplace's method applied to certain integrals, and the Euler-Maclaurin summation formula) covers the case of one or two hard walls at the boundary, or inside, of the droplet for which the soft edge formulas are shown to no longer hold. Use of this to study the large $N$ expansion of the cylinder partition function requires knowledge of the same expansion in the annular case. This is not available in the case of two hard walls inside the droplet, which we treat instead via a direct analysis. In distinction to the case of the annular partition function, the large $N$ expansion of the logarithm of the cylinder partition function is found to always have no term proportional to $\log N$, independent of the presence of hard walls.
Comments45 pages
