Random normal matrices in the almost-circular regime

Abstract

We study random normal matrix models whose eigenvalues tend to be distributed within a narrow "band" around the unit circle of width proportional to 1 n , where n is the size of matrices. For general radially symmetric potentials with various boundary conditions, we derive the scaling limits of the correlation functions, some of which appear in the previous literature notably in the context of almost-Hermitian random matrices. We also obtain that fluctuations of the maximal and minimal modulus of the ensembles follow the Gumbel or exponential law depending on the boundary conditions.

Exponential moments for disk counting statistics of random normal matrices in the critical regime

“…The Mittag-Leffler ensemble has been widely studied over the years, see e.g. [15,2,25,3,4,9] for various universality and finite-n results. We also refer the interested reader to [28] for more background on two-dimensional point processes.…”

On the almost-circular symplectic induced Ginibre ensemble

“…For the other case |p| = 1, p = −1, 1, we find that the Pfaffian structure of (1.1) simplifies in the limit and becomes determinantal, see (1.8)-(1.9). The limiting correlation kernel K C is given by (1.10) and already appeared in [29,27,28,7,16,14,20], but the way it arises in this work, namely as the large N limit of the kernel of a Pfaffian point process, is new to our knowledge. In Theorem 1.1 (c), we study a transition between the two families of correlation functions associated with κ R and K C .…”

“…Planar ensembles with thin droplets were introduced in the works [29,27,28] on the complex elliptic Ginibre ensemble and were recently studied in [14,20] for more general random normal matrix models. The work [14] treats universality questions for general bandlimited point processes using Ward's equation, while [20] deals with almost-circular ensembles associated with general radially symmetric potentials. In the earlier works [29,27,28,14,20], the considered point processes are determinantal.…”

“…The work [14] treats universality questions for general bandlimited point processes using Ward's equation, while [20] deals with almost-circular ensembles associated with general radially symmetric potentials. In the earlier works [29,27,28,14,20], the considered point processes are determinantal. A major difference with our case is that (1.1) is Pfaffian.…”

Exponential moments for disk counting statistics at the hard edge of random normal matrices

“…Coulomb gas ensembles in the presence of a hard wall have previously been considered in the literature, but so far the focus has been on large gap probabilities (or partition functions) [50,47,53,4,5,3,1,48,29] and on the local statistics [78,64,71]. We refer to [11,70,12,23,51,59] for studies of local droplets and local statistics near soft/hard edges.…”