05052nam 2200565 450 991047889190332120211029211415.01-4704-4821-1(CKB)4100000007133851(MiAaPQ)EBC5571104(PPN)231946376(Au-PeEL)EBL5571104(OCoLC)1054216569(EXLCZ)99410000000713385120181203d2018 uy 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierOn mesoscopic equilibrium for linear statistics in Dyson's Brownian motion /Maurice Duits, Kurt JohanssonProvidence, Rhode Island :American Mathematical Society,[2018]©20181 online resource (130 pages)Memoirs of the American Mathematical Society ;Volume 255, Number 12221-4704-2964-0 Includes bibliographical references.Cover -- 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< <1/2 -- 7.5. Estimating ^{ _{ }^{\eps}}_{ } for 0< <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 -- 9.5. Proof of Theorem 2.6: the general case -- Appendix -- Bibliography -- Back Cover.In 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.Memoirs of the American Mathematical Society ;Volume 255, Number 1222.Stochastic processesStochastic differential equationsMesoscopic phenomena (Physics)Brownian motion processesElectronic books.Stochastic processes.Stochastic differential equations.Mesoscopic phenomena (Physics)Brownian motion processes.519.2/33Duits Maurice964789Johansson Kurt1960-MiAaPQMiAaPQMiAaPQBOOK9910478891903321On mesoscopic equilibrium for linear statistics in Dyson's Brownian motion2441898UNINA01143nam0 22002651i 450 UON0049590620231205105353.6620190624d1919 |0itac50 baitaIT|||| |||||ˆGli ‰arazzi dei Gonzaga restituiti dall'AustriaAlessandro LuzioBergamoOfficine dell'Istituto Italiano d'Arti Grafiche191938 p. [20 c. di tav] in gran parte ill.35 cm.ARAZZISec. XVIUONC095193FIITBergamoUONL000244746.3945DIPINTI, ARAZZI, TAPPEZZERIE. Italia22LUZIOAlessandroUONV132855136925Officine dell'istituto italiano d'Arti graficheUONV283515650ITSOL20251212RICASIBA - SISTEMA BIBLIOTECARIO DI ATENEO E ARCHIVIO STORICOUONSIUON00495906SIBA - SISTEMA BIBLIOTECARIO DI ATENEO E ARCHIVIO STORICOSI CAT 248 SI 30589 7 Arazzi dei Gonzaga restituiti dall'Austria521596UNIOR