Asymptotic expansion of the partition function for $β$-ensembles with complex potentials
In this work we establish under certain hypotheses the $N \to +\infty$ asymptotic expansion of integrals of the form $$\mathcal{Z}_{N,Γ}[V] \, = \, \int_{Γ^N} \prod_{ a < b}^{N}(z_a - z_b)^β\, \prod_{k=1}^{N} \mathrm{e}^{ - N βV(z_k) } \, \mathrm{d}\mathbf{z}$$ where $V \in \mathbb{C}[X]$, $β\in 2 \mathbb{N}^*$ is an even integer and $Γ\subset \mathbb{C}$ is an unbounded contour such that the integral converges. For even degree, real valued $V$s and when $Γ= \mathbb{R}$, it is well known that the large-$N$ expansion is characterised by an equilibrium measure corresponding to the minimiser of an appropriate energy functional. This method bears a structural resemblance with the Laplace method. By contrast, in the complex valued setting we are considering, the analysis structurally resembles the classical steepest-descent method, and involves finding a critical point \textit{and} a steepest descent curve, the latter being a deformation of the original integration contour. More precisely, one minimises a curve-dependent energy functional with respect to measures on the curve and then maximises the energy over an appropriate space of curves. Our analysis deals with the one-cut regime of the associated equilibrium measure. We establish the existence of an all order asymptotic expansion for $\ln \mathcal{Z}_{N,Γ}[V]$ and explicitly identify the first few terms.