Hole Probabilities for Finite and Infinite Ginibre Ensembles:

Abstract

Abstract. We study the hole probabilities of the infinite Ginibre ensemble X∞, a determinantal point process on the complex plane with the kernelwith 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 thatwhere ∅ is the empty set andis the space of all compactly supported probability measures with support in U c . Using potential theory, we give an explicit formula for R U , the minimum possible energy of a probability measure compactly supported on U c under logarithmic potential with a quadratic external field. Moreover, we calculate R U explicitly for some special sets like annulus, cardioid, ellipse, equilateral triangle and half disk.

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.…”

“…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].…”

Point Processes, Hole Events, and Large Deviations: Random Complex Zeros and Coulomb Gases

“…The proof of the lower bound is similar to the proof of the lower bound for the hole probability in [1], and we will only sketch the idea. We again choose t = R −C 1 for some sufficiently large C 1 > 0, and scale the points as follows w = 1 R z.…”