Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\boldsymbol{X}_1, \cdots, \boldsymbol{X}_{m_n}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> be independent n × n complex Ginibre ensembles and Z1, ⋯, Zn be the eigenvalues of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\prod_{j=1}^{m_{n}} \boldsymbol{X}_{j}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <munderover> <mo>∏</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <msub> <mi>m</mi> <mrow> <mi>n</mi> </mrow> </msub> </mrow> </munderover> <msub> <mi mathvariant="bold-italic">X</mi> <mrow> <mi>j</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. Suppose <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\lim\limits_{n\to\infty}m_n=+\infty$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <munder> <mo form="prefix">lim</mo> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>m</mi> <mi>n</mi> </msub> <mo>=</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </math></EquationSource> </InlineEquation>, we obtain large and moderate deviations for max1≤i≤n log ∣Zi∣.

Deviation probabilities for spectral radius of products of complex Ginibre ensembles

期刊论文
发表日期：

2025年12月22日

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Yutao Ma,
Chaoyang Song

摘要

Let $\boldsymbol{X}_1, \cdots, \boldsymbol{X}_{m_n}$ $P (M)$ be independent n × n complex Ginibre ensembles and Z₁, ⋯, Z_n be the eigenvalues of $\prod_{j=1}^{m_{n}} \boldsymbol{X}_{j}$ $\prod_{j = 1}^{m_{n}} X_{j}$ . Suppose $\lim\limits_{n\to\infty}m_n=+\infty$ $lim_{n \to \infty} m_{n} = + \infty$ , we obtain large and moderate deviations for max_1≤i≤n log ∣Z_i∣.