Deformation of semi-circle law for the correlated time series and Phase transition

We study the eigenvalue of the Wigner random matrix, which is created from a time series with temporal correlation. We observe the deformation of the semi-circle law which is similar to the eigenvalue distribution of the Wigner-L\`{e}vy matrix. The distribution has a longer tail and a higher peak than the semi-circle law. In the absence of correlation, the eigenvalue distribution of the Wigner random matrix is known as the semi-circle law in the large $N$ limit. When there is a temporal correlation, the eigenvalue distribution converges to the deformed semi-circle law which has a longer tail and a higher peak than the semi-circle law. When we created the Wigner matrix using financial time series, we test the normal i.i.d. using the Wigner matrix. We observe the difference from the semi-circle law for FX time series. The difference from the semi-circle law is explained by the temporal correlation. Here, we discuss the moments of distribution and convergence to the deformed semi-circle law with a temporal correlation. We discuss the phase transition and compare to the Marchenko-Pastur distribution(MPD) case.