Hole probabilities for finite and infinite Ginibre ensembles

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We study the hole probabilities of the infinite Ginibre ensemble X∞, a determinantal point process on the complex plane with the kernel K(z,w)= 1πez w-1/2|z|²-1/2|w|² with respect to the Lebesgue measure on the complex plane. Let U be an open subset of open unit disk D and X∞(rU) denote the number of points of X∞ that fall in rU. Then, under some conditions on U, we show that ᵣ→ ∞1/r⁴[X∞(rU)=0]=R-Rᵤ, where is the empty set and Rᵤ:=μ∈ ₚ₍ᵤᶜ₎{ 1/|z-w|dμ(z)dμ(w)+∫ |z|²dμ(z) }, P(Uᶜ) is the space of all compactly supported probability measures with support in Uᶜ. Using potential theory, we give an explicit formula for Rᵤ, the minimum possible energy of a probability measure compactly supported on Uᶜ under logarithmic potential with a quadratic external field. Moreover, we calculate Rᵤ explicitly for some special sets like annulus, cardioid, ellipse, equilateral triangle and half disk.