Geodesic
A note on pair-dependent linear statistics with a slowly increasing variance
Teoretičeskaâ i matematičeskaâ fizika, Tome 205 (2020) no. 3, pp. 502-512 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We prove Gaussian fluctuations for pair-counting statistics of the form $\Sigma_{1\le i\ne j\le N}f(\theta_i-\theta_j)$ for the circular unitary ensemble of random matrices in the large-$N$ limit under the condition that the variance increases slowly as $N$ increases.
Mots-clés : random matrix
Keywords: circular unitary ensemble, pair-counting statistics, central limit theorem. 
@article{TMF_2020_205_3_a8,
     author = {A. Aguirre and A. B. Soshnikov},
     title = {A~note on pair-dependent linear statistics with a~slowly increasing variance},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {502--512},
     year = {2020},
     volume = {205},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2020_205_3_a8/}
}
TY  - JOUR
AU  - A. Aguirre
AU  - A. B. Soshnikov
TI  - A note on pair-dependent linear statistics with a slowly increasing variance
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2020
SP  - 502
EP  - 512
VL  - 205
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2020_205_3_a8/
LA  - ru
ID  - TMF_2020_205_3_a8
ER  - 
%0 Journal Article
%A A. Aguirre
%A A. B. Soshnikov
%T A note on pair-dependent linear statistics with a slowly increasing variance
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2020
%P 502-512
%V 205
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2020_205_3_a8/
%G ru
%F TMF_2020_205_3_a8
A. Aguirre; A. B. Soshnikov. A note on pair-dependent linear statistics with a slowly increasing variance. Teoretičeskaâ i matematičeskaâ fizika, Tome 205 (2020) no. 3, pp. 502-512. http://geodesic.mathdoc.fr/item/TMF_2020_205_3_a8/

[1] F. J. Dyson, “Statistical theory of the energy levels of complex systems. I”, J. Math. Phys., 3:1 (1962), 140–156 ; “Statistical theory of the energy levels of complex systems. II”, 157–165 ; “Statistical theory of the energy levels of complex systems. III”, 166–175 | DOI | MR | DOI | MR | DOI | MR

[2] F. J. Dyson, “Correlations between the eigenvalues of a random matrix”, Commun. Math. Phys., 19:3 (1970), 235–250 | DOI | MR

[3] M. L. Meta, Sluchainye matritsy, MTsNMO, M., 2012 | MR

[4] R. Killip, I. Nenciu, “Matrix models for circular ensembles”, Internat. Math. Res. Not., 2004:50 (2004), 2665–2701 | DOI | MR

[5] A. Aguirre, A. Soshnikov, J. Sumpter, “Pair dependent linear statistics for C$\beta$E”, Random Matrices Theory Appl. (to appear) , arXiv: DOI: https://doi.org/10.1142/S20103263215003501912.07110

[6] H. L. Montgomery, “The pair correlation of zeros of the zeta function”, Analytic Number Theory, Proceedings of Symposia in Pure Mathematics, 24, ed. H. G. Diamond, AMS, Providence, RI, 1973, 181–193 | DOI | MR

[7] H. L. Montgomery, “Distribution of the zeros of the Riemann zeta function”, Proceedings of the International Congress of Mathematicians (Vancouver, BC, 1974), v. 1, Canad. Math. Congress, Montreal, Que, 1975, 379–381 | MR

[8] N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, 27, Cambridge Univ. Press, Cambridge, 1987 | DOI | MR

[9] K. Johansson, “On fluctuations of eigenvalues of random Hermitian matrices”, Duke Math. J., 91:1 (1988), 151–204 | DOI | MR

[10] P. Diaconis, S. N. Evans, “Linear functionals of eigenvalues of random matrices”, Trans. Amer. Math. Soc., 353:7 (2001), 2615–2633 | DOI | MR

[11] A. Soshnikov, “The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities”, Ann. Probab., 28:3 (2000), 1353–1370 | DOI | MR

[12] F. Bekerman, A. Lodhia, “Mesoscopic central limit theorem for general $\beta$-ensembles”, Ann. Inst. Henri Poincaré Probab. Stat., 54:4 (2018), 1917–1938 | DOI | MR

[13] G. Lambert, Mesoscopic central limit theorem for the circular beta-ensembles and applications, arXiv: 1902.06611

[14] P. Diaconis, M. Shahshhani, “On the eigenvalues of random matrices”, J. Appl. Probab., 31 (1994), 49–62 | DOI | MR

[15] K. Johansson, “On random matrices from the compact classical groups”, Ann. Math. Ser. 2, 145:3 (1997), 519–545 | DOI | MR

[16] T. H. Baker, P. J. Forrester, “Finite-$N$ fluctuation formulas for random matrices”, J. Stat. Phys., 88:5–6 (1997), 1371–1386 | DOI | MR

[17] A. Soshnikov, “Level spacings distribution for large random matrices: Gaussian fluctuations”, Ann. Math. Ser. 2, 148:2 (1998), 573–617 | DOI | MR

[18] C. P. Hughes, J. P. Keating, N. O'Connell, “On the characteristic polynomial of a random unitary matrix”, Commun. Math. Phys., 220:2 (2001), 429–451 | DOI | MR

[19] E. S. Meckes, M. W. Meckes, “Self-similarity in the circular unitary ensemble”, Discrete Anal., 2016 (2016), 9, 14 pp., arXiv: 1507.05876 | DOI | MR

[20] E. Paquette, O. Zeitouni, “The maximum of the CUE field”, Internat. Math. Res. Not., 2018:16 (2018), 5028–5119 | DOI | MR

[21] C. Webb, “Linear statistics of the circular $\beta$-ensemble, Stein's method, and circular Dyson Brownian motion”, Electron. J. Probab., 21 (2016), 25, 16pp pp. | DOI | MR

[22] N. S. Witte, P. J. Forrester, “Moments of the Gaussian $\beta$ ensembles and the large-$N$ expansion of the densities”, J. Math. Phys., 55:8 (2014), 083302, 34 pp., arXiv: 1310.8498 | DOI | MR

[23] T. Jiang, S. Matsumoto, “Moments of traces of circular beta-ensembles”, Ann. Probab., 43:6 (2015), 3279–3336 | DOI | MR