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

• We consider the multivariate moment generating function of the disk counting statistics of a model Mittag-Leffler ensemble in the presence of a hard wall. Let n be the number of points. We focus on two regimes: (a) the “hard edge regime” where all disk boundaries are at a distance of order n1 from the hard wall, and (b) the “semi-hard edge regime” where all disk boundaries are at a distance of order √1n from the hard wall. As n → + ∞, we prove that the moment generating function enjoys asymptotics of the form (Equation presented) In both cases, we determine the constants C1;:::; C4 explicitly. We also derive precise asymptotic formulas for all joint cumulants of the disk counting function, and establish several central limit theorems. Surprisingly, and in contrast to the “bulk”, “soft edge”, and “semi-hard edge” regimes, the second and higher order cumulants of the disk counting function in the “hard edge” regime are proportional to n and not to √n.

Ämnesord

NATURVETENSKAP  -- Matematik -- Sannolikhetsteori och statistik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Probability Theory and Statistics (hsv//eng)

NATURVETENSKAP  -- Matematik -- Matematisk analys (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Mathematical Analysis (hsv//eng)

Nyckelord

asymptotic analysis

Moment generating functions

random matrix theory

asymptotic analysis

Moment generating functions

random matrix theory

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