In random matrix theory, Let $A$ be a random $n \times n$ matrix whose entries i.i.d with expectation 0 and variance 1, let F be the LSD of $A$ , $F$ will be uniform distribution over the unit disk. The marginal distribution of $F$ equal to a non-standard semicircle law, which is the LSD of a wigner matrix $B$. Is there any relationship between circle law and semicircle law ? That is, if we have known one of the two law, can we prove another one by means of matrix analysis?
This is a good, interesting question, with lots of possible paths to follow. When you write "circle law", I am guessing that you mean "full circle law" (see Mueller, 2000).
The semicircle law (Wigner), full circle law, quarter circle law, deformed quarter circle law and others are covered in
R.R. Mueller, Random matrix theory. Tutorial overview, 2000:
http://lthiwww.epfl.ch/~leveque/Matrix/muller.pdf
For more about this, see
S. Verdu, Eigenvalues and singular values of random matrices, 2006:
http://www.ims.nus.edu.sg/Programs/randommatrix/files/sverdu_p.pdf
The full circle law is given on page 22 and quarter circle law appears on page 26.
and also check
T. Rogers, New results on the spectral density of random matrices, PhD thesis, King's College, London, 2010:
http://people.bath.ac.uk/ma3tcr/tim_rogers_thesis.pdf
A proof of Wigner's semicircle law is given on page 16.
This is a good, interesting question, with lots of possible paths to follow. When you write "circle law", I am guessing that you mean "full circle law" (see Mueller, 2000).
The semicircle law (Wigner), full circle law, quarter circle law, deformed quarter circle law and others are covered in
R.R. Mueller, Random matrix theory. Tutorial overview, 2000:
http://lthiwww.epfl.ch/~leveque/Matrix/muller.pdf
For more about this, see
S. Verdu, Eigenvalues and singular values of random matrices, 2006:
http://www.ims.nus.edu.sg/Programs/randommatrix/files/sverdu_p.pdf
The full circle law is given on page 22 and quarter circle law appears on page 26.
and also check
T. Rogers, New results on the spectral density of random matrices, PhD thesis, King's College, London, 2010:
http://people.bath.ac.uk/ma3tcr/tim_rogers_thesis.pdf
A proof of Wigner's semicircle law is given on page 16.
The reply of James Peters is to the point and includes many references and explanations.
Let me take a different angle on things - both the circle law and the semi-circle law are some sort of large-number-theorems, namely capture some universal average behaviour of a large collection of random numbers (in this case organized as a matrix). The eigenvalues of those random matrices converge to one of the above-mentioned laws under quite general conditions (i.e., even if the individual entries are not Gaussian or uniformly distributed - any distribution with finite variance usually would do the job - Tim Rogers' PhD thesis indeed discusses this for example).
The main difference between the laws is that in the case of the circle law the eigenvalues are complex while in Wigner's semi-circle law the eigenvalues are real - and the reason is simple. In the first case the random matrices are non-hermitian while in the second they are hermitian (or even symmetric).
Obtaining one from the other is non-trivial, but there are techniques that take the non-hermitian case, and create from them an ensemble of larger hermitian matrices (double the size) and then use results for those hermitian ones (such as the semi-circle law) to infer results for the non-hermitian ones.
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