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

ArXiv ID: 2508.07192 “View on arXiv”

Authors: Masato Hisakado, Takuya Kaneko

Abstract

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-Lèvy 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.

Keywords: Wigner random matrix, temporal correlation, deformed semi-circle law, Marchenko-Pastur distribution, eigenvalue distribution, FX

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail
• Why: The paper uses advanced random matrix theory, deriving moments and phase transitions via heavy LaTeX formulas, while applying the theory to real FX data, including statistical tests and convergence analysis for practical implementation.

  flowchart TD
    A["Research Goal<br>Investigate the effect of temporal correlation<br>on the eigenvalue distribution of Wigner random matrices<br>derived from time series"] --> B["Data & Inputs<br>Financial Time Series (FX)<br>Standard Wigner Matrix (i.i.d.)<br>Wigner-Lévy Matrix (Reference)"]
    
    B --> C["Methodology<br>Construct Wigner Random Matrix<br>from time series data"]
    
    C --> D["Computational Process<br>Calculate Eigenvalues<br>Compute Moments of Distribution<br>Check Convergence in Large N limit"]
    
    D --> E["Key Findings<br>Deformed Semi-Circle Law<br>(Longer tail, Higher peak)"]
    
    E --> F["Outcomes<br>1. Correlation explains deviation from semi-circle law<br>2. Phase transition observed vs Marchenko-Pastur distribution<br>3. Validation on FX data"]