Gaussian Unitary Ensembles with Pole Singularities Near the Soft Edge and a System of Coupled Painlevé XXXIV Equations

In this paper, we study the singularly perturbed Gaussian unitary ensembles defined by the measure

(1/C_n)e⁻n^tr V ⁽M^;λ^,t^→) dM, 

over the space of n × n Hermitian matrices M, where V (x; λ,t^→) := 2x² + Σ^2m_k=1 t_k (x − λ)^−k with t^→ = (t₁, t₂,...,t_2m) ∈ R^2m−1 × (0, ∞), in the multiple scaling limit, where λ → 1 together with t^→ → 0^→ as n → ∞ at appropriate related rates. We obtain the asymptotics of the partition function, which is described explicitly in terms of an integral involving a smooth solution to a new coupled Painlevé system generalizing the Painlevé XXXIV equation. The large n limit of the correlation kernel is also derived, which leads to a new universal class built out of the Ψ-function associated with the coupled Painlevé system.