We consider N classical particles interacting via the Coulomb potential in spatial dimension d and in the presence of an external trap, at equilibrium at inverse temperature β. In the large N limit, the particles are confined within a droplet of finite size. We study smooth linear statistics, i.e. the fluctuations of sums of the form LN=∑i=1Nf(xi) , where x _i ’s are the positions of the particles and where f(xi) is a sufficiently regular function. There exists at present standard results for the first and second moments of LN in the large N limit, as well as associated Central Limit Theorems in general dimension and for a wide class of confining potentials. Here we obtain explicit expressions for the higher order cumulants of LN at large N, when the function f(x)=f(|x|) and the confining potential are both rotationnally invariant. A remarkable feature of our results is that these higher cumulants depend only on the value of f′(|x|) and its higher order derivatives evaluated exactly at the boundary of the droplet, which in this case is a d-dimensional sphere. In the particular two-dimensional case d = 2 at the special value β = 2, a connection to the Ginibre ensemble allows us to derive these results in an alternative way using the tools of determinantal point processes. Finally we also obtain the large deviation form of the full probability distribution function of LN .

中文翻译：

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库仑气体的线性统计：高阶累积量

我们认为氮经典粒子通过空间维度中的库仑势相互作用d并在存在外部陷阱的情况下，在逆温下达到平衡β。在大氮极限，粒子被限制在有限尺寸的液滴内。我们研究平滑线性统计，即形式之和的波动 L氮=Σ我=1氮F（X我） ， 在哪里X _我 是粒子的位置，其中 F（X我） 是一个足够正则的函数。目前存在一阶矩和二阶矩的标准结果 L氮 在大氮极限，以及一般维度和各种限制势的相关中心极限定理。这里我们获得了高阶累积量的显式表达式 L氮 在逃氮，当函数 F（X）=F（|X|） 并且约束势都是旋转不变的。我们结果的一个显着特征是这些较高的累积量仅取决于 F′（|X|） 及其高阶导数在液滴边界处精确评估，在本例中是d维度球体。在特定的二维情况下d = 2 在特殊值β = 2，与 Ginibre 系综的连接允许我们使用行列式点过程工具以另一种方式得出这些结果。最后我们还得到了全概率分布函数的大偏差形式 L氮 。