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Carlo Beenakker
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It is important to distinguish results that apply only in the limit of a large matrix size $N$, from results that apply for small $N$ as well. The theory of random matrices addresses both large-$N$ properties as well as small-$N$ properties, and there are physical applications in both regimes.

I understand from your question that your interest is in large-$N$ properties of the GOE. Then "bulk" versus edge refers to the support $(-W,W)$ of the Wigner semicircle, rescaled such that $W={\cal O}(1)$. The transition from bulk to edge region is at a separation of order $N^{-2/3}$ from $\pm W$. The deviation of the eigenvalue density from the Wigner semicircle in the edge region is described by the Tracy-Widom law.

But do keep in mind that the theory of random matrices is not restricted to the large-$N$ regime. For example, chaotic scattering from a billiard with an $N\times N$ scatteringtransmission matrix is described by the circular ensembles for any $N$, even as small as $N=1,2,3,\ldots$. In that case there is no notion of bulk versus edge, but there are universal properties that can be measured in experiments., such as the (Here is an$T^{-1/2}(1-T)^{-1/2}$ distribution of the transmission probability overview.)$T\in(0,1)$ for $N=1$.

It is important to distinguish results that apply only in the limit of a large matrix size $N$, from results that apply for small $N$ as well. The theory of random matrices addresses both large-$N$ properties as well as small-$N$ properties, and there are physical applications in both regimes.

I understand from your question that your interest is in large-$N$ properties of the GOE. Then "bulk" versus edge refers to the support $(-W,W)$ of the Wigner semicircle, rescaled such that $W={\cal O}(1)$. The transition from bulk to edge region is at a separation of order $N^{-2/3}$ from $\pm W$. The deviation of the eigenvalue density from the Wigner semicircle in the edge region is described by the Tracy-Widom law.

But do keep in mind that the theory of random matrices is not restricted to the large-$N$ regime. For example, chaotic scattering from a billiard with an $N\times N$ scattering matrix is described by the circular ensembles for any $N$, even as small as $N=1,2,3,\ldots$. In that case there is no notion of bulk versus edge, but there are universal properties that can be measured in experiments. (Here is an overview.)

It is important to distinguish results that apply only in the limit of a large matrix size $N$, from results that apply for small $N$ as well. The theory of random matrices addresses both large-$N$ properties as well as small-$N$ properties, and there are physical applications in both regimes.

I understand from your question that your interest is in large-$N$ properties of the GOE. Then "bulk" versus edge refers to the support $(-W,W)$ of the Wigner semicircle, rescaled such that $W={\cal O}(1)$. The transition from bulk to edge region is at a separation of order $N^{-2/3}$ from $\pm W$. The deviation of the eigenvalue density from the Wigner semicircle in the edge region is described by the Tracy-Widom law.

But do keep in mind that the theory of random matrices is not restricted to the large-$N$ regime. For example, chaotic scattering from a billiard with an $N\times N$ transmission matrix is described by the circular ensembles for any $N$, even as small as $N=1,2,3,\ldots$. In that case there is no notion of bulk versus edge, but there are universal properties that can be measured in experiments, such as the $T^{-1/2}(1-T)^{-1/2}$ distribution of the transmission probability $T\in(0,1)$ for $N=1$.

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Carlo Beenakker
  • 188.1k
  • 18
  • 448
  • 651

It is important to distinguish results that apply only in the limit of a large matrix size $N$, from results that apply for small $N$ as well. The theory of random matrices addresses both large-$N$ properties as well as small-$N$ properties, and there are physical applications in both regimes.

I understand from your question that your interest is in large-$N$ properties of the GOE. Then "bulk" versus edge refers to the support $(-W,W)$ of the Wigner semicircle, rescaled such that $W={\cal O}(1)$. The transition from bulk to edge region is at a separation of order $N^{-2/3}$ from $\pm W$. The deviation of the eigenvalue density from the Wigner semicircle in the edge region is described by the Tracy-Widom law.

But do keep in mind that the theory of random matrices is not restricted to the large-$N$ regime. For example, chaotic scattering from a billiard with an $N\times N$ scattering matrix is described by the circular ensembles for any $N$, even as small as $N=1,2,3,\ldots$. In that case there is no notion of bulk versus edge, but there are universal properties that can be measured in experiments. (Here is an overview.)