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

We study the partition function $$ Z_n = \int_{\mathbb{C}^n } \prod_{1 \le j<k \le n} |z_{j}-z_{k}|^{2} \prod_{j=1}^{n} |z_j|^{2c}\, e^{-n V(z_{j})}\frac{d^{2}z_{j}}π, $$ 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 Deaño 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.