2017/01/05 by J. A. Mendez-Bermudez, J A Méndez-Bermúdez, Guilherme Ferraz de Arruda +3
Computer Science · Mathematics · Physics and Astronomy · #Distribution (mathematics) #Eigenfunction #Eigenvalues and eigenvectors #Matrix (chemical analysis) #Quantum Information and Cryptography #Random Matrices and Applications #Random matrix #Scaling #Spectral Theory in Mathematical Physics #Spectral power distribution #Spectral properties #cond-mat.dis-nn
paper · pdf · doi:10.1088/1751-8121/aa9509
6 pages, 3 figures. arXiv admin note: text overlap with arXiv:1611.06695
arxiv created 2017/01/05 · openalex created_date 2017/01/26 · openalex publication_date 2017/10/20 · arxiv updated 2017/12/06 · openalex updated_date 2026/08/06
Abstract We demonstrate that the normalized localization length β of the eigenfunctions of diluted (sparse) banded random matrices follows the scaling law <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mstyle> </mml:math> . The scaling parameter of the model is defined as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>∝</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mi>b</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> <mml:mi mathvariant="normal">f</mml:mi> <mml:mi mathvariant="normal">f</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mrow> <mml:mspace width="0pt"/> </mml:mrow> <mml:mi>δ</mml:mi> </mml:msup> </mml:mstyle> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mi>b</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> <mml:mi mathvariant="normal">f</mml:mi> <mml:mi mathvariant="normal">f</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mstyle> </mml:math> is the average number of non-zero elements per matrix row, N is the matrix size, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>δ</mml:mi> <mml:mo>∼</mml:mo> <mml:mn>1</mml:mn> </mml:mstyle> </mml:math> . Additionally, we show that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mstyle> </mml:math> also scales the spectral properties of the model (up to certain sparsity) characterized by the spacing distribution of eigenvalues.