Deviation probabilities for extremal eigenvalues of large Chiral non-Hermitian random matrices

Consider the chiral non-Hermitian random matrix ensemble with parameters n and v , and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>ζ</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>i</m:mi> <m:mo>≤</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> {(ζᵢ)1≤ i≤ n} be its n eigenvalues with positive x -coordinate. In this paper, we establish deviation probabilities and moderate deviation probabilities for the spectral radius <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mfrac> <m:mi>n</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mi>v</m:mi> </m:mrow> </m:mfrac> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:msub> <m:mi>max</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>i</m:mi> <m:mo>≤</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>⁡</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:msub> <m:mi>ζ</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> </m:math> {(n/n+v)1/2max1≤ i≤ n|ζᵢ|²} , as well as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mfrac> <m:mi>n</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mi>v</m:mi> </m:mrow> </m:mfrac> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:msub> <m:mi>min</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>i</m:mi> <m:mo>≤</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>⁡</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:msub> <m:mi>ζ</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> </m:math> {(n/n+v)1/2min1≤ i≤ n|ζᵢ|²} .