Hole probabilities for finite and infinite Ginibre ensembles

Hole probabilities for $\beta $-ensembles and determinantal point processes in the complex plane

“…It turns out that the decay constants are explicit in terms of the minimum energies of the sets, depending on external fields. As a corollary of these results, we get the hole probability results for finite Ginibre ensembles, proved in [AR16].…”

“…We derive the decay constants explicitly in terms of the minimum energies of the sets, with the external fields |z| α . In particular, α = 2 gives the hole probability results for infinite Ginibre ensemble, proved in [AR16].…”

“…In this thesis we prove the similar asymptotic results of hole probabilities for X (α) ∞ . The essential ideas of the proofs are borrowed from the paper [AR16]. As a consequence of these results, for α = 2, we get the hole probabilities for the infinite Ginibre ensemble.…”

“…In this section we calculate the equilibrium measures and the minimum energies for some sets with quadratic external fields. The following theorem has been proved in [AR16], a particular case (for g (|z|) = |z| 2 ) of Theorem 2.6.…”

“…The equilibrium measure µ is uniform measure on D, i.e., d µ(z) = 1 π d m(z) on D and the minimum energy is R (2) = 3 4 . Proof of this particular case can be found in [AR16].…”