eBook ISBN:  9781470448219 
Product Code:  MEMO/255/1222.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $46.80 
eBook ISBN:  9781470448219 
Product Code:  MEMO/255/1222.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $46.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 255; 2018; 118 pp
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 (sineuniversality 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.

Table of Contents

Chapters

1. Introduction

2. Statement of results

3. Proof of Theorem

4. Proof of Theorem

5. Asymptotic analysis of $K_n$ and $R_n$

6. Proof of Proposition

7. Proof of Lemma

8. Random initial points

9. Proof of Theorem : the general case

Appendix


Additional Material

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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 (sineuniversality 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.

Chapters

1. Introduction

2. Statement of results

3. Proof of Theorem

4. Proof of Theorem

5. Asymptotic analysis of $K_n$ and $R_n$

6. Proof of Proposition

7. Proof of Lemma

8. Random initial points

9. Proof of Theorem : the general case

Appendix