Saddle points in the chaotic analytic function and Ginibre characteristic polynomial

Comparison is made between the distribution of saddle points in the chaotic analytic function and in the characteristic polynomials of the Ginibre ensemble. Realising the logarithmic derivative of these infinite polynomials as the electric field of a distribution of coulombic charges at the zeros, a simple mean-field electrostatic argument shows that the density of saddles minus zeros falls off as π −1 |z| −4 from the origin. This behaviour is expected to be general for finite or infinite polynomials with zeros uniformly randomly distributed in the complex plane, and which repel quadratically. It is well-known that there are several similarities between the distributions of zeros of the ensemble of random polynomials which tend to the chaotic analytic function 1991 chapter 15), both in the finite and infinite cases. To be more precise, for N large and possibly ∞, we compare the zeros of the chaotic analytic function polynomials (caf polynomials) f N,caf (z) = N n=0 a n z n √ n! , (1) where the a n are independent identically distributed complex circular gaussian random variables, and the eigenvalues of matrices in the Ginibre ensemble, defined to be N × N matrices with entries independent identically distributed complex circular gaussian random variables. The Ginibre analogue to equation (1) is the characteristic polynomial f N,Gin (z). The zeros of the two f N share the following properties, as discussed in the above references: • they are uniformly randomly distributed, with density σ = 1/π, within a disk centred on the origin of the complex plane, with radius √ N , and boundary Gauss-smoothed; • within this disk, the distribution of zeros is statistically invariant to translation and rotation; • the statistical properties of the zeros at a fixed radius r 0 do not change as N increases, when N ≫ r 2 0 ;