Extremal eigenvalue correlations in the GUE minor process and a law of fractional logarithm

• Autores: Elliot Paquette, Ofer Zeitouni
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 6, 1, 2017, págs. 4112-4166
• Idioma: inglés
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• Resumen

  • Let λ(N)λ(N) be the largest eigenvalue of the N×NN×N GUE matrix which is the NNth element of the GUE minor process, rescaled to converge to the standard Tracy–Widom distribution. We consider the sequence {λ(N)}N≥1{λ(N)}N≥1 and prove a law of fractional logarithm for the lim suplim sup:

    lim supN→∞λ(N)(logN)2/3=(14)2/3almost surely.

    lim supN→∞λ(N)(log⁡N)2/3=(14)2/3almost surely.

    For the lim inflim inf, we prove the weaker result that there are constants c1,c2>0c1,c2>0 so that −c1≤lim infN→∞λ(N)(logN)1/3≤−c2almost surely.

    −c1≤lim infN→∞λ(N)(log⁡N)1/3≤−c2almost surely.

    We conjecture that in fact, c1=c2=41/3c1=c2=41/3.