One-eigenvalue large-deviation rate for non-Hermitian invariant ensembles

Establish that, for non-Hermitian matrix ensembles with density proportional to exp{-N Tr W(M,M*)} dM, connected limiting spectral support, and an analytically continuable bulk electrostatic potential, the one-eigenvalue large-deviation rate outside the support equals the difference between the analytically continued bulk branch and the true exterior logarithmic potential, namely Ψ(z)=V_bulk(z, z̄)-U_μ(z).

Background

The paper studies non-Hermitian ensembles for which an explicit Coulomb-gas eigenvalue density is generally unavailable. In the Coulomb-gas setting, the cost of moving one eigenvalue outside the limiting spectral support is obtained from the confining potential minus the logarithmic potential generated by the equilibrium measure.

The authors conjecture that an analogous effective-potential formula remains valid for generic non-Hermitian ensembles of the form dP_N(M) proportional to exp{-N Tr W(M,M*)} dM, provided the bulk branch of the electrostatic potential admits an analytic continuation to the exterior. This would extend the standard one-particle large-deviation mechanism beyond ensembles with explicitly known eigenvalue laws.