Spectral Gap and Edge Universality of Dense Random Regular Graphs
Abstract Let $${\mathcal {A}}$$ be the adjacency matrix of a random d-regular graph on N vertices, and we denote its eigenvalues by $$\lambda _1\geqslant \lambda _2\cdots \geqslant \lambda _{N}$$. For $$N^{2/3+o(1)}\leqslant d\leqslant N/2$$, we prove optimal rigidity estimates of the extreme eigenv...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
He, Yukun [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2024 |
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| Anmerkung: |
© The Author(s) 2024 |
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| Übergeordnetes Werk: |
Enthalten in: Communications in mathematical physics - Springer Berlin Heidelberg, 1965, 405(2024), 8 vom: 23. Juli |
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| Übergeordnetes Werk: |
volume:405 ; year:2024 ; number:8 ; day:23 ; month:07 |
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| DOI / URN: |
10.1007/s00220-024-05063-x |
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| Katalog-ID: |
SPR056689594 |
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| 520 | |a Abstract Let $${\mathcal {A}}$$ be the adjacency matrix of a random d-regular graph on N vertices, and we denote its eigenvalues by $$\lambda _1\geqslant \lambda _2\cdots \geqslant \lambda _{N}$$. For $$N^{2/3+o(1)}\leqslant d\leqslant N/2$$, we prove optimal rigidity estimates of the extreme eigenvalues of $${\mathcal {A}}$$, which in particular imply that max{|λN|,λ2}<2d-1$$\begin{aligned} \max \{|\lambda _N|,\lambda _2\} <2\sqrt{d-1} \end{aligned}$$with very high probability. In the same regime of d, we also show that N2/3(λ2+d/Nd(N-d)/N-2)⟶dTW1,$$\begin{aligned} N^{2/3}\bigg (\frac{\lambda _2+d/N}{\sqrt{d(N-d)/N}}-2\bigg ) \overset{d}{\longrightarrow }\ \textrm{TW}_1, \end{aligned}$$where $$\textrm{TW}_1$$ is the Tracy–Widom distribution for GOE; analogue results also hold for other non-trivial extreme eigenvalues. | ||
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10.1007/s00220-024-05063-x doi (DE-627)SPR056689594 (SPR)s00220-024-05063-x-e DE-627 ger DE-627 rakwb eng 530 510 VZ 33.06 bkl 31.80 bkl He, Yukun verfasserin (orcid)0000-0002-8559-5263 aut Spectral Gap and Edge Universality of Dense Random Regular Graphs 2024 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2024 Abstract Let $${\mathcal {A}}$$ be the adjacency matrix of a random d-regular graph on N vertices, and we denote its eigenvalues by $$\lambda _1\geqslant \lambda _2\cdots \geqslant \lambda _{N}$$. For $$N^{2/3+o(1)}\leqslant d\leqslant N/2$$, we prove optimal rigidity estimates of the extreme eigenvalues of $${\mathcal {A}}$$, which in particular imply that max{|λN|,λ2}<2d-1$$\begin{aligned} \max \{|\lambda _N|,\lambda _2\} <2\sqrt{d-1} \end{aligned}$$with very high probability. In the same regime of d, we also show that N2/3(λ2+d/Nd(N-d)/N-2)⟶dTW1,$$\begin{aligned} N^{2/3}\bigg (\frac{\lambda _2+d/N}{\sqrt{d(N-d)/N}}-2\bigg ) \overset{d}{\longrightarrow }\ \textrm{TW}_1, \end{aligned}$$where $$\textrm{TW}_1$$ is the Tracy–Widom distribution for GOE; analogue results also hold for other non-trivial extreme eigenvalues. Enthalten in Communications in mathematical physics Springer Berlin Heidelberg, 1965 405(2024), 8 vom: 23. Juli (DE-627)253721628 (DE-600)1458931-X 1432-0916 nnns volume:405 year:2024 number:8 day:23 month:07 https://dx.doi.org/10.1007/s00220-024-05063-x X:SPRINGER Resolving-System kostenfrei Volltext SYSFLAG_0 GBV_SPRINGER SSG-OPC-MAT GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 33.06 VZ 31.80 VZ AR 405 2024 8 23 07 |
| spelling |
10.1007/s00220-024-05063-x doi (DE-627)SPR056689594 (SPR)s00220-024-05063-x-e DE-627 ger DE-627 rakwb eng 530 510 VZ 33.06 bkl 31.80 bkl He, Yukun verfasserin (orcid)0000-0002-8559-5263 aut Spectral Gap and Edge Universality of Dense Random Regular Graphs 2024 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2024 Abstract Let $${\mathcal {A}}$$ be the adjacency matrix of a random d-regular graph on N vertices, and we denote its eigenvalues by $$\lambda _1\geqslant \lambda _2\cdots \geqslant \lambda _{N}$$. For $$N^{2/3+o(1)}\leqslant d\leqslant N/2$$, we prove optimal rigidity estimates of the extreme eigenvalues of $${\mathcal {A}}$$, which in particular imply that max{|λN|,λ2}<2d-1$$\begin{aligned} \max \{|\lambda _N|,\lambda _2\} <2\sqrt{d-1} \end{aligned}$$with very high probability. In the same regime of d, we also show that N2/3(λ2+d/Nd(N-d)/N-2)⟶dTW1,$$\begin{aligned} N^{2/3}\bigg (\frac{\lambda _2+d/N}{\sqrt{d(N-d)/N}}-2\bigg ) \overset{d}{\longrightarrow }\ \textrm{TW}_1, \end{aligned}$$where $$\textrm{TW}_1$$ is the Tracy–Widom distribution for GOE; analogue results also hold for other non-trivial extreme eigenvalues. Enthalten in Communications in mathematical physics Springer Berlin Heidelberg, 1965 405(2024), 8 vom: 23. Juli (DE-627)253721628 (DE-600)1458931-X 1432-0916 nnns volume:405 year:2024 number:8 day:23 month:07 https://dx.doi.org/10.1007/s00220-024-05063-x X:SPRINGER Resolving-System kostenfrei Volltext SYSFLAG_0 GBV_SPRINGER SSG-OPC-MAT GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 33.06 VZ 31.80 VZ AR 405 2024 8 23 07 |
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10.1007/s00220-024-05063-x doi (DE-627)SPR056689594 (SPR)s00220-024-05063-x-e DE-627 ger DE-627 rakwb eng 530 510 VZ 33.06 bkl 31.80 bkl He, Yukun verfasserin (orcid)0000-0002-8559-5263 aut Spectral Gap and Edge Universality of Dense Random Regular Graphs 2024 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2024 Abstract Let $${\mathcal {A}}$$ be the adjacency matrix of a random d-regular graph on N vertices, and we denote its eigenvalues by $$\lambda _1\geqslant \lambda _2\cdots \geqslant \lambda _{N}$$. For $$N^{2/3+o(1)}\leqslant d\leqslant N/2$$, we prove optimal rigidity estimates of the extreme eigenvalues of $${\mathcal {A}}$$, which in particular imply that max{|λN|,λ2}<2d-1$$\begin{aligned} \max \{|\lambda _N|,\lambda _2\} <2\sqrt{d-1} \end{aligned}$$with very high probability. In the same regime of d, we also show that N2/3(λ2+d/Nd(N-d)/N-2)⟶dTW1,$$\begin{aligned} N^{2/3}\bigg (\frac{\lambda _2+d/N}{\sqrt{d(N-d)/N}}-2\bigg ) \overset{d}{\longrightarrow }\ \textrm{TW}_1, \end{aligned}$$where $$\textrm{TW}_1$$ is the Tracy–Widom distribution for GOE; analogue results also hold for other non-trivial extreme eigenvalues. Enthalten in Communications in mathematical physics Springer Berlin Heidelberg, 1965 405(2024), 8 vom: 23. Juli (DE-627)253721628 (DE-600)1458931-X 1432-0916 nnns volume:405 year:2024 number:8 day:23 month:07 https://dx.doi.org/10.1007/s00220-024-05063-x X:SPRINGER Resolving-System kostenfrei Volltext SYSFLAG_0 GBV_SPRINGER SSG-OPC-MAT GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 33.06 VZ 31.80 VZ AR 405 2024 8 23 07 |
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10.1007/s00220-024-05063-x doi (DE-627)SPR056689594 (SPR)s00220-024-05063-x-e DE-627 ger DE-627 rakwb eng 530 510 VZ 33.06 bkl 31.80 bkl He, Yukun verfasserin (orcid)0000-0002-8559-5263 aut Spectral Gap and Edge Universality of Dense Random Regular Graphs 2024 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2024 Abstract Let $${\mathcal {A}}$$ be the adjacency matrix of a random d-regular graph on N vertices, and we denote its eigenvalues by $$\lambda _1\geqslant \lambda _2\cdots \geqslant \lambda _{N}$$. For $$N^{2/3+o(1)}\leqslant d\leqslant N/2$$, we prove optimal rigidity estimates of the extreme eigenvalues of $${\mathcal {A}}$$, which in particular imply that max{|λN|,λ2}<2d-1$$\begin{aligned} \max \{|\lambda _N|,\lambda _2\} <2\sqrt{d-1} \end{aligned}$$with very high probability. In the same regime of d, we also show that N2/3(λ2+d/Nd(N-d)/N-2)⟶dTW1,$$\begin{aligned} N^{2/3}\bigg (\frac{\lambda _2+d/N}{\sqrt{d(N-d)/N}}-2\bigg ) \overset{d}{\longrightarrow }\ \textrm{TW}_1, \end{aligned}$$where $$\textrm{TW}_1$$ is the Tracy–Widom distribution for GOE; analogue results also hold for other non-trivial extreme eigenvalues. Enthalten in Communications in mathematical physics Springer Berlin Heidelberg, 1965 405(2024), 8 vom: 23. Juli (DE-627)253721628 (DE-600)1458931-X 1432-0916 nnns volume:405 year:2024 number:8 day:23 month:07 https://dx.doi.org/10.1007/s00220-024-05063-x X:SPRINGER Resolving-System kostenfrei Volltext SYSFLAG_0 GBV_SPRINGER SSG-OPC-MAT GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 33.06 VZ 31.80 VZ AR 405 2024 8 23 07 |
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10.1007/s00220-024-05063-x doi (DE-627)SPR056689594 (SPR)s00220-024-05063-x-e DE-627 ger DE-627 rakwb eng 530 510 VZ 33.06 bkl 31.80 bkl He, Yukun verfasserin (orcid)0000-0002-8559-5263 aut Spectral Gap and Edge Universality of Dense Random Regular Graphs 2024 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2024 Abstract Let $${\mathcal {A}}$$ be the adjacency matrix of a random d-regular graph on N vertices, and we denote its eigenvalues by $$\lambda _1\geqslant \lambda _2\cdots \geqslant \lambda _{N}$$. For $$N^{2/3+o(1)}\leqslant d\leqslant N/2$$, we prove optimal rigidity estimates of the extreme eigenvalues of $${\mathcal {A}}$$, which in particular imply that max{|λN|,λ2}<2d-1$$\begin{aligned} \max \{|\lambda _N|,\lambda _2\} <2\sqrt{d-1} \end{aligned}$$with very high probability. In the same regime of d, we also show that N2/3(λ2+d/Nd(N-d)/N-2)⟶dTW1,$$\begin{aligned} N^{2/3}\bigg (\frac{\lambda _2+d/N}{\sqrt{d(N-d)/N}}-2\bigg ) \overset{d}{\longrightarrow }\ \textrm{TW}_1, \end{aligned}$$where $$\textrm{TW}_1$$ is the Tracy–Widom distribution for GOE; analogue results also hold for other non-trivial extreme eigenvalues. Enthalten in Communications in mathematical physics Springer Berlin Heidelberg, 1965 405(2024), 8 vom: 23. Juli (DE-627)253721628 (DE-600)1458931-X 1432-0916 nnns volume:405 year:2024 number:8 day:23 month:07 https://dx.doi.org/10.1007/s00220-024-05063-x X:SPRINGER Resolving-System kostenfrei Volltext SYSFLAG_0 GBV_SPRINGER SSG-OPC-MAT GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 33.06 VZ 31.80 VZ AR 405 2024 8 23 07 |
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He, Yukun |
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He, Yukun ddc 530 bkl 33.06 bkl 31.80 Spectral Gap and Edge Universality of Dense Random Regular Graphs |
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530 510 VZ 33.06 bkl 31.80 bkl Spectral Gap and Edge Universality of Dense Random Regular Graphs |
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Spectral Gap and Edge Universality of Dense Random Regular Graphs |
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Communications in mathematical physics |
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spectral gap and edge universality of dense random regular graphs |
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Spectral Gap and Edge Universality of Dense Random Regular Graphs |
| abstract |
Abstract Let $${\mathcal {A}}$$ be the adjacency matrix of a random d-regular graph on N vertices, and we denote its eigenvalues by $$\lambda _1\geqslant \lambda _2\cdots \geqslant \lambda _{N}$$. For $$N^{2/3+o(1)}\leqslant d\leqslant N/2$$, we prove optimal rigidity estimates of the extreme eigenvalues of $${\mathcal {A}}$$, which in particular imply that max{|λN|,λ2}<2d-1$$\begin{aligned} \max \{|\lambda _N|,\lambda _2\} <2\sqrt{d-1} \end{aligned}$$with very high probability. In the same regime of d, we also show that N2/3(λ2+d/Nd(N-d)/N-2)⟶dTW1,$$\begin{aligned} N^{2/3}\bigg (\frac{\lambda _2+d/N}{\sqrt{d(N-d)/N}}-2\bigg ) \overset{d}{\longrightarrow }\ \textrm{TW}_1, \end{aligned}$$where $$\textrm{TW}_1$$ is the Tracy–Widom distribution for GOE; analogue results also hold for other non-trivial extreme eigenvalues. © The Author(s) 2024 |
| abstractGer |
Abstract Let $${\mathcal {A}}$$ be the adjacency matrix of a random d-regular graph on N vertices, and we denote its eigenvalues by $$\lambda _1\geqslant \lambda _2\cdots \geqslant \lambda _{N}$$. For $$N^{2/3+o(1)}\leqslant d\leqslant N/2$$, we prove optimal rigidity estimates of the extreme eigenvalues of $${\mathcal {A}}$$, which in particular imply that max{|λN|,λ2}<2d-1$$\begin{aligned} \max \{|\lambda _N|,\lambda _2\} <2\sqrt{d-1} \end{aligned}$$with very high probability. In the same regime of d, we also show that N2/3(λ2+d/Nd(N-d)/N-2)⟶dTW1,$$\begin{aligned} N^{2/3}\bigg (\frac{\lambda _2+d/N}{\sqrt{d(N-d)/N}}-2\bigg ) \overset{d}{\longrightarrow }\ \textrm{TW}_1, \end{aligned}$$where $$\textrm{TW}_1$$ is the Tracy–Widom distribution for GOE; analogue results also hold for other non-trivial extreme eigenvalues. © The Author(s) 2024 |
| abstract_unstemmed |
Abstract Let $${\mathcal {A}}$$ be the adjacency matrix of a random d-regular graph on N vertices, and we denote its eigenvalues by $$\lambda _1\geqslant \lambda _2\cdots \geqslant \lambda _{N}$$. For $$N^{2/3+o(1)}\leqslant d\leqslant N/2$$, we prove optimal rigidity estimates of the extreme eigenvalues of $${\mathcal {A}}$$, which in particular imply that max{|λN|,λ2}<2d-1$$\begin{aligned} \max \{|\lambda _N|,\lambda _2\} <2\sqrt{d-1} \end{aligned}$$with very high probability. In the same regime of d, we also show that N2/3(λ2+d/Nd(N-d)/N-2)⟶dTW1,$$\begin{aligned} N^{2/3}\bigg (\frac{\lambda _2+d/N}{\sqrt{d(N-d)/N}}-2\bigg ) \overset{d}{\longrightarrow }\ \textrm{TW}_1, \end{aligned}$$where $$\textrm{TW}_1$$ is the Tracy–Widom distribution for GOE; analogue results also hold for other non-trivial extreme eigenvalues. © The Author(s) 2024 |
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Spectral Gap and Edge Universality of Dense Random Regular Graphs |
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https://dx.doi.org/10.1007/s00220-024-05063-x |
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2024-08-22T04:49:32.088Z |
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