A smooth transition towards a Tracy-Widom distribution for the largest eigenvalue of interacting k-body fermionic embedded Gaussian ensembles
Por:
Carro, Ernesto, Benet, Luis, Pérez Castillo I.
Publicada:
1 abr 2023
Resumen:
In spite of its simplicity, the central limit theorem captures one of
the most outstanding phenomena in mathematical physics, that of
universality. While this classical result is well understood, it is
still not very clear what hap -pens to this universal behavior when the
random variables become correlated. A fruitful mathematically laboratory
to investigate the rising of new universal properties is offered by the
set of eigenvalues of random matrices. In this regard, a lot of work has
been done using the standard random matrix ensembles and focusing on the
distribution of extreme eigenvalues. In this case, the distribution of
the largest or smallest eigenvalue departs from the
Fisher-Tippett-Gnedenko theorem yielding the celebrated Tracy-Widom
distribution. One may wonder, yet again, how robust this new universal
behavior captured by the Tracy-Widom distribution is when the
correlation among eigenvalues changes. Few answers have been provided to
this poignant question, and our intention in the present work is to
contribute to this interesting unexplored territory. Thus, we study
numerically the probability distribution for the normalized largest
eigenvalue of the interacting k-body fermionic orthogonal and unitary
embedded Gaussian ensembles in the diluted limit. We find a smooth
transition from a slightly asymmetric Gaussian-like distribution, for
small k/m, to the Tracy-Widom distribution as k/m ? 1, where k is the
rank of the interaction and m is the number of fermions. Correlations at
the edge of the spectrum are stronger for small values of k/m and are
independent of the number of particles considered. Our results indicate
that subtle correlations toward the edge of the spectrum distinguish the
statistical properties of the spectrum of interacting many-body systems
in the dilute limit, from those expected for the standard random matrix
ensembles.
Filiaciones:
Carro, Ernesto:
Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Av. Universidad s/n, Col. Chamilpa Morelos, Cuernavaca, CP 62210, Mexico
Univ Nacl Autonoma Mexico, Inst Ciencias Fis, Ave Univ S-N, Cuernavaca 62210, Morelos, Mexico
Benet, Luis:
Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Av. Universidad s/n, Col. Chamilpa Morelos, Cuernavaca, CP 62210, Mexico
Univ Nacl Autonoma Mexico, Inst Ciencias Fis, Ave Univ S-N, Cuernavaca 62210, Morelos, Mexico
Pérez Castillo I.:
Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Ciudad de Mexico, 09340, Mexico
Univ Autonoma Metropolitana Iztapalapa, Dept Fis, San Rafael Atlixco 186, Mexico City 09340, Mexico
Green Submitted, hybrid, Hybrid Gold
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