Large gap asymptotics on annuli in the random normal matrix model
Christophe Charlier
Abstract
Abstract We consider a two-dimensional determinantal point process arising in the random normal matrix model and which is a two-parameter generalization of the complex Ginibre point process. In this paper, we prove that the probability that no points lie on any number of annuli centered at 0 satisfies large n asymptotics of the form $$\begin{aligned} \exp \Bigg ( C_{1} n^{2} + C_{2} n \log n + C_{3} n + C_{4} \sqrt{n} + C_{5}\log n + C_{6} + {\mathcal {F}}_{n} + \mathcal {O}\Big ( n^{-\frac{1}{12}}\Big )\Bigg ), \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mo>exp</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> </mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mi>n</mml:mi> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:msqrt> <mml:mi>n</mml:mi> </mml:msqrt> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>5</mml:mn> </mml:msub> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>6</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> </mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>12</mml:mn> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> where n is the number of points of the process. We determine the constants $$C_{1},\ldots ,C_{6}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>6</mml:mn> </mml:msub> </mml:mrow> </mml:math> explicitly, as well as the oscillatory term $${\mathcal {F}}_{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> which is of order 1. We also allow one annulus to be a disk, and one annulus to be unbounded. For the complex Ginibre point process, we improve on the best known results: (i) when the hole region is a disk, only $$C_{1},\ldots ,C_{4}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> </mml:math> were previously known, (ii) when the hole region is an unbounded annulus, only $$C_{1},C_{2},C_{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> </mml:math> were previously known, and (iii) when the hole region is a regular annulus in the bulk, only $$C_{1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> was previously known. For general values of our parameters, even $$C_{1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> is new. A main discovery of this work is that $${\mathcal {F}}_{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> is given in terms of the Jacobi theta function. As far as we know this is the first time this function appears in a large gap problem of a two-dimensional point process.