标题： 有条件GinUE的自由能展开和LUE最小特征值的大偏差 显示英文标题

标题： Free energy expansions of a conditional GinUE and large deviations of the smallest eigenvalue of the LUE

摘要： 我们考虑一个规模为$N$的平面库仑气体系综，其逆温度为$\beta=2$，外部势能为$Q(z)=|z|^2-2c \log|z-a|$，其中 $c>0$和$a \in \mathbb{C}$。 等价地，该模型可以实现为大小为$(c+1) N \times (c+1) N$的复Ginibre矩阵的$N$个特征值，条件是具有多重性为$cN$的确定性特征值$a$。 根据$c$和$a$值的不同，液滴显示出相变：在临界后阶段是二连通的，在临界前阶段是一连通的。 在两种情况下，我们都推导出自由能的精确的大$N$展开式直到$O(1)$项，提供了一个非径向对称的例子，证实了为一般平面库仑气体系综提出的Zabrodin-Wiegmann猜想。 由此，我们的结果提供了复Ginibre矩阵特征多项式的矩的渐近行为，其中幂的阶数为$O(N)$。 此外，通过结合对偶公式，我们获得了拉盖尔幺正系综最小特征值的确切大偏差概率。 我们的证明基于利用部分Schlesinger变换的平面正交多项式的精细Riemann-Hilbert分析。 显示英文摘要

摘要： We consider a planar Coulomb gas ensemble of size $N$ with the inverse temperature $\beta=2$ and external potential $Q(z)=|z|^2-2c \log|z-a|$, where $c>0$ and $a \in \mathbb{C}$. Equivalently, this model can be realised as $N$ eigenvalues of the complex Ginibre matrix of size $(c+1) N \times (c+1) N$ conditioned to have deterministic eigenvalue $a$ with multiplicity $cN$. Depending on the values of $c$ and $a$, the droplet reveals a phase transition: it is doubly connected in the post-critical regime and simply connected in the pre-critical regime. In both regimes, we derive precise large-$N$ expansions of the free energy up to the $O(1)$ term, providing a non-radially symmetric example that confirms the Zabrodin-Wiegmann conjecture made for general planar Coulomb gas ensembles. As a consequence, our results provide asymptotic behaviours of moments of the characteristic polynomial of the complex Ginibre matrix, where the powers are of order $O(N)$. Furthermore, by combining with a duality formula, we obtain precise large deviation probabilities of the smallest eigenvalue of the Laguerre unitary ensemble. Our proof is based on a refined Riemann-Hilbert analysis for planar orthogonal polynomials using the partial Schlesinger transform.