Random matrices with log-range correlations, and log-Sobolev inequalities
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Bibliographic record
Abstract
Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> be a symmetric <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>×</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> random matrix whose <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mi>N</mml:mi> </mml:msqrt> </mml:math> -scaled entries are uniformly square integrable. We prove that if the entries of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> can be partitioned into independent subsets each of size <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mo form="prefix">log</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , then the empirical eigenvalue distribution of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> , minus its mean, converges weakly to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>0</mml:mn> </mml:math> in probability; hence if the averaged empirical eigenvalue distribution converges to a law, the empirical spectral distribution converges to this limit law weakly in probability. If the entries are bounded, the convergence is almost sure; if the entries are Gaussian, we prove almost sure convergence with larger blocks of size <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>N</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>/</mml:mo> <mml:mo form="prefix">log</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . This significantly extends the best previously known results on convergence of eigenvalues for matrices with correlated entries, where the partition subsets are blocks and of size <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . We also prove the strongest known convergence results for eigenvalues of band matrices. We prove these results by developing a new log-Sobolev inequality which generalizes the second author’s introduction of mollified log-Sobolev inequalities: we show that if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="bold">Y</mml:mi> </mml:math> is a bounded random vector and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="bold">Z</mml:mi> </mml:math> is a standard normal random vector independent from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="bold">Y</mml:mi> </mml:math> , then the law of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="bold">Y</mml:mi> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi mathvariant="bold">Z</mml:mi> </mml:mrow> </mml:math> satisfies a log-Sobolev inequality for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , and we give bounds on the optimal log-Sobolev constant.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
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| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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