# Universal shocks in the Wishart random-matrix ensemble.

@article{Blaizot2012UniversalSI, title={Universal shocks in the Wishart random-matrix ensemble.}, author={J. P. Blaizot and Maciej A. Nowak and Piotr Warchol}, journal={Physical review. E, Statistical, nonlinear, and soft matter physics}, year={2012}, volume={87 5}, pages={ 052134 } }

We show that the derivative of the logarithm of the average characteristic polynomial of a diffusing Wishart matrix obeys an exact partial differential equation valid for an arbitrary value of N, the size of the matrix. In the large N limit, this equation generalizes the simple inviscid Burgers equation that has been obtained earlier for Hermitian or unitary matrices. The solution, through the method of characteristics, presents singularities that we relate to the precursors of shock formation…

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## References

SHOWING 1-10 OF 56 REFERENCES

### Universal shocks in random matrix theory.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2010

The edge of the spectrum of eigenvalues is related to the shock that naturally appears in the Burgers equation for appropriate initial conditions, thereby suggesting a connection between the well-known microscopic universality of random matrix theory and the universal properties of the solution of the Burger equation in the vicinity of a shock.

### Large-Nc confinement and turbulence.

- PhysicsPhysical review letters
- 2008

It is suggested that the transition that occurs at large N_{c} in the eigenvalue distribution of a Wilson loop may have a turbulent origin, and the appearance of a shock in the spectral flow of the Wilson loop eigenvalues is demonstrated.

### Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems

- Mathematics
- 2004

As an extension of the theory of Dyson’s Brownian motion models for the standard Gaussian random-matrix ensembles, we report a systematic study of Hermitian matrix-valued processes and their…

### Universal random matrix correlations of ratios of characteristic polynomials at the spectral edges

- Mathematics
- 2003

### Multiplying unitary random matrices—universality and spectral properties

- Mathematics
- 2003

In this paper, we calculate, in the large N limit, the eigenvalue density of an infinite product of random unitary matrices, each of them generated by a random Hermitian matrix. This is equivalent to…

### Characteristic Polynomials of Random Matrices

- Mathematics
- 2000

Abstract: Number theorists have studied extensively the connections between the distribution of zeros of the Riemann ζ-function, and of some generalizations, with the statistics of the eigenvalues of…

### Nonintersecting Brownian interfaces and Wishart random matrices.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2009

It is shown that, for a large system and with an appropriate choice of the external confining potential, the joint distribution of the heights of the N nonintersecting interfaces at a fixed point on the substrate can be mapped to the joint Distribution of the eigenvalues of a Wishart matrix of size N with complex entries, thus providing a physical realization of the Wishart Matrix.

### Exponential Ensemble for Random Matrices

- Mathematics, Physics
- 1965

Using the ideas of information theory, it is pointed out that the Gaussian ensemble for random Hermitian matrices can be characterized as the ``most random'' ensemble of these matrices, and the definition of the exponential ensemble is led in a natural way.

### Why is a shock not a caustic? The higher-order Stokes phenomenon and smoothed shock formation

- Mathematics
- 2007

The formation of shocks in waves of advance in nonlinear partial differential equations is a well-explored problem and has been studied using many different techniques. In this paper we demonstrate…