标题： 平面库仑气体的异常自由能展开：多组分和共形奇点 显示英文标题

标题： Anomalous free energy expansions of planar Coulomb gases: multi-component and conformal singularity

摘要： 我们研究分割函数$$ Z_n = \int_{\mathbb{C}^n } \prod_{1 \le j<k \le n} |z_{j}-z_{k}|^{2} \prod_{j=1}^{n} |z_j|^{2c}\, e^{-n V(z_{j})}\frac{d^{2}z_{j}}{\pi}, $$，其中$c>-1$和$$ V(z)= |z|^{2d}-t(z^{d}+\overline{z}^{d}), \qquad t >0, \, d \in \mathbb{N}. $$。相关的液滴揭示了一个拓扑相变：对于$t > 1/\sqrt{d}$，它由$d$个连通分支组成；而当$t < 1/\sqrt{d}$时，它变为单连通并包含原点，在那里出现共形奇异性。 在两种情况下，我们建立了渐近展开式$$ \log Z_n = C_1 n^2 + C_2 n \log n + C_3 n + C_4 \log n + C_5 + O(n^{-1}), $$作为$n \to \infty$，并显式地推导出所有系数。 在多组分情形$t > 1/\sqrt{d}$中，常数项$C_5$展现出一种振荡行为，该行为取决于$n$对$d$的模同余类。 特别是，在特殊情形 $c = 0$且 $n$被 $d$整除的情况下，我们的结果验证了Deaño和Simm的猜想。 相反，在共形奇点区域 $t < 1/\sqrt{d}$，振荡行为消失，而在 $C_4$中出现额外贡献，超出了Jancovici等人猜想的范围。 作为特殊情况 $d = 1$，我们的结果给出了复Ginibre系综特征多项式的矩的渐近展开式，其指数为有限值。 在体区 regime 中，我们进一步推导了 $1/N$的幂级数展开，从而对Webb和Wong的结果中的误差项进行了精确评估。 显示英文摘要

摘要： We study the partition function $$ Z_n = \int_{\mathbb{C}^n } \prod_{1 \le j<k \le n} |z_{j}-z_{k}|^{2} \prod_{j=1}^{n} |z_j|^{2c}\, e^{-n V(z_{j})}\frac{d^{2}z_{j}}{\pi}, $$ where $c>-1$ and $$ V(z)= |z|^{2d}-t(z^{d}+\overline{z}^{d}), \qquad t >0, \, d \in \mathbb{N}. $$ The associated droplet reveals a topological phase transition: for $t > 1/\sqrt{d}$, it consists of $d$ connected components; whereas for $t < 1/\sqrt{d}$, it becomes simply connected and contains the origin, where a conformal singularity arises. In both regimes, we establish the asymptotic expansion $$ \log Z_n = C_1 n^2 + C_2 n \log n + C_3 n + C_4 \log n + C_5 + O(n^{-1}), $$ as $n \to \infty$, and derive all coefficients explicitly. In the multi-component regime $t > 1/\sqrt{d}$, the constant term $C_5$ exhibits an oscillatory behaviour that depends on the congruence class of $n$ modulo $d$. In particular, in the special case $c = 0$ with $n$ divisible by $d$, our result confirms a conjecture of Dea\~no and Simm. In contrast, in the conformal singularity regime $t < 1/\sqrt{d}$, the oscillatory behaviour disappears, while additional contributions in $C_4$ arise beyond the scope of the conjecture of Jancovici et al. As a special case $d = 1$, our result yields the asymptotic expansion for the moments of the characteristic polynomial of the complex Ginibre ensemble with finite exponent. In the bulk regime, we further derive the full expansion in powers of $1/N$, thereby providing a precise evaluation of the error term in the result of Webb and Wong.