Crystallization of random matrix orbits
Three operations on eigenvalues of real/complex/quaternion (corresponding to β=1,2,4) matrices, obtained from cutting out principal corners, adding, and multiplying matrices, can be extrapolated to general values of β>0 through associated special functions. We show that the β→∞ limit for these o...
| Main Authors: | , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
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Oxford University Press (OUP)
2020
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| Online Access: | https://hdl.handle.net/1721.1/124436 |
| _version_ | 1826204378518781952 |
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| author | Gorin, Vadim Marcus, Adam W. |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Gorin, Vadim Marcus, Adam W. |
| author_sort | Gorin, Vadim |
| collection | MIT |
| description | Three operations on eigenvalues of real/complex/quaternion (corresponding to β=1,2,4) matrices, obtained from cutting out principal corners, adding, and multiplying matrices, can be extrapolated to general values of β>0 through associated special functions. We show that the β→∞ limit for these operations leads to the finite free projection, additive convolution, and multiplicative convolution, respectively. The limit is the most transparent for cutting out the corners, where the joint distribution of the eigenvalues of principal corners of a uniformly-random general β self-adjoint matrix with fixed eigenvalues is known as the β-corners process. We show that as β→∞ these eigenvalues crystallize on an irregular lattice consisting of the roots of derivatives of a single polynomial. In the second order, we observe a version of the discrete Gaussian Free Field put on top of this lattice, which provides a new explanation as to why the (continuous) Gaussian Free Field governs the global asymptotics of random matrix ensembles. ©2020 |
| first_indexed | 2024-09-23T12:53:46Z |
| format | Article |
| id | mit-1721.1/124436 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T12:53:46Z |
| publishDate | 2020 |
| publisher | Oxford University Press (OUP) |
| record_format | dspace |
| spelling | mit-1721.1/1244362022-09-28T10:47:24Z Crystallization of random matrix orbits Gorin, Vadim Marcus, Adam W. Massachusetts Institute of Technology. Department of Mathematics Three operations on eigenvalues of real/complex/quaternion (corresponding to β=1,2,4) matrices, obtained from cutting out principal corners, adding, and multiplying matrices, can be extrapolated to general values of β>0 through associated special functions. We show that the β→∞ limit for these operations leads to the finite free projection, additive convolution, and multiplicative convolution, respectively. The limit is the most transparent for cutting out the corners, where the joint distribution of the eigenvalues of principal corners of a uniformly-random general β self-adjoint matrix with fixed eigenvalues is known as the β-corners process. We show that as β→∞ these eigenvalues crystallize on an irregular lattice consisting of the roots of derivatives of a single polynomial. In the second order, we observe a version of the discrete Gaussian Free Field put on top of this lattice, which provides a new explanation as to why the (continuous) Gaussian Free Field governs the global asymptotics of random matrix ensembles. ©2020 NSF (Grant DMS-1407562) NSF (Grant DMS-1664619) 2020-03-30T20:44:37Z 2020-03-30T20:44:37Z 2020-02 2017-11 2019-11-12T17:52:15Z Article http://purl.org/eprint/type/JournalArticle 1687-0247 1073-7928 https://hdl.handle.net/1721.1/124436 Gorin, Vadim, and Adam W. Marcus, "Crystallization of random matrix orbits." International mathematics research notices 2020, 3 (February 2020): p. 883-913 doi 10.1093/imrn/rny052 ©2020 Author(s) en 10.1093/IMRN/RNY052 International mathematics research notices Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Oxford University Press (OUP) arXiv |
| spellingShingle | Gorin, Vadim Marcus, Adam W. Crystallization of random matrix orbits |
| title | Crystallization of random matrix orbits |
| title_full | Crystallization of random matrix orbits |
| title_fullStr | Crystallization of random matrix orbits |
| title_full_unstemmed | Crystallization of random matrix orbits |
| title_short | Crystallization of random matrix orbits |
| title_sort | crystallization of random matrix orbits |
| url | https://hdl.handle.net/1721.1/124436 |
| work_keys_str_mv | AT gorinvadim crystallizationofrandommatrixorbits AT marcusadamw crystallizationofrandommatrixorbits |