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006			m o d |
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008			240724s2018 xx o ||||0 eng d
020			_a9781470448219 _q(electronic bk.)
020			_z9781470429645
035			_a(MiAaPQ)EBC5571104
035			_a(Au-PeEL)EBL5571104
035			_a(OCoLC)1054216569
040			_aMiAaPQ _beng _erda _epn _cMiAaPQ _dMiAaPQ
050		4	_aQA274.75 .D858 2018
082	0		_a530.475
100	1		_aDuits, Maurice.
245	1	0	_aOn Mesoscopic Equilibrium for Linear Statistics in Dyson's Brownian Motion.
250			_a1st ed.
264		1	_aProvidence : _bAmerican Mathematical Society, _c2018.
264		4	_c©2018.
300			_a1 online resource (130 pages)
336			_atext _btxt _2rdacontent
337			_acomputer _bc _2rdamedia
338			_aonline resource _bcr _2rdacarrier
490	1		_aMemoirs of the American Mathematical Society Series ; _vv.255
505	0		_aCover -- Title page -- Chapter 1. Introduction -- Chapter 2. Statement of results -- 2.1. Assumptions on ⱼ⁽ⁿ⁾ -- 2.2. Deterministic initial points -- 2.3. Concentration inequalities -- 2.4. Random initial points -- 2.5. Further remarks -- 2.6. Overview of the rest of the paper -- Chapter 3. Proof of Theorem 2.1 -- 3.1. Determinantal strucure -- 3.2. Asymptotic results for _{ } and _{ }^{ } -- 3.3. Proof of Theorem 2.1 -- Chapter 4. Proof of Theorem 2.3 -- 4.1. Overview of the proof -- 4.2. The loop equations -- 4.3. Loop equations on the mesoscopic scale -- 4.4. Proof of Theorem 2.3 -- Chapter 5. Asymptotic analysis of _{ } and _{ } -- 5.1. Integrable form of _{ } -- 5.2. The functions ℰⱼ -- 5.3. Saddle points -- 5.4. Deforming the contours -- 5.5. Asymptotics for ⱼ and ⱼ -- 5.6. Proof of Lemma 3.2 -- 5.7. Asymptotics for _{ }( , ) -- 5.8. Asymptotics for ^{ }_{ } -- Chapter 6. Proof of Proposition 2.4 -- 6.1. Preliminaries -- 6.2. A first concentration inequality -- 6.3. Proof of Poposition 6.2 -- 6.4. A concentration inequality using the logaritmic Sobolev inequality -- 6.5. Proof of Proposition 2.4 -- 6.6. One more concentration inequality -- Chapter 7. Proof of Lemma 4.3 -- 7.1. Preliminaries -- 7.2. Estimating _{ }^{ _{ }^{\eps}} -- 7.3. Estimating _{ }^{ _{ }^{\eps}} -- 7.4. Estimating ^{ _{ }^{\eps}}_{ } for 0&lt -- &lt -- 1/2 -- 7.5. Estimating ^{ _{ }^{\eps}}_{ } for 0&lt -- &lt -- 1 -- Chapter 8. Random initial points -- 8.1. Preliminary lemmas -- 8.2. Regularity of the initial points -- 8.3. Smoothening the test function -- 8.4. Approximating _{ }( ) -- 8.5. Proof of Theorem 2.5, and Theorem 2.6 with the assumption ₀( )≠0 -- Chapter 9. Proof of Theorem 2.6: the general case -- 9.1. Smoothening of the test function -- 9.2. Change of variables -- 9.3. Expansion into moments -- 9.4. Proof of Proposition 9.1.
505	8		_a9.5. Proof of Theorem 2.6: the general case -- Appendix -- Bibliography -- Back Cover.
520			_aIn this paper the authors study mesoscopic fluctuations for Dyson's Brownian motion with \beta =2. Dyson showed that the Gaussian Unitary Ensemble (GUE) is the invariant measure for this stochastic evolution and conjectured that, when starting from a generic configuration of initial points, the time that is needed for the GUE statistics to become dominant depends on the scale we look at: The microscopic correlations arrive at the equilibrium regime sooner than the macrosopic correlations. The authors investigate the transition on the intermediate, i.e. mesoscopic, scales. The time scales that they consider are such that the system is already in microscopic equilibrium (sine-universality for the local correlations), but have not yet reached equilibrium at the macrosopic scale. The authors describe the transition to equilibrium on all mesoscopic scales by means of Central Limit Theorems for linear statistics with sufficiently smooth test functions. They consider two situations: deterministic initial points and randomly chosen initial points. In the random situation, they obtain a transition from the classical Central Limit Theorem for independent random variables to the one for the GUE.
588			_aDescription based on publisher supplied metadata and other sources.
590			_aElectronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
650		0	_aBrownian motion processes.
655		4	_aElectronic books.
700	1		_aJohansson, Kurt.
776	0	8	_iPrint version: _aDuits, Maurice _tOn Mesoscopic Equilibrium for Linear Statistics in Dyson's Brownian Motion _dProvidence : American Mathematical Society,c2018 _z9781470429645
797	2		_aProQuest (Firm)
830		0	_aMemoirs of the American Mathematical Society Series
856	4	0	_uhttps://ebookcentral.proquest.com/lib/orpp/detail.action?docID=5571104 _zClick to View
999			_c6264 _d6264