Universality Conjecture and Results for a Model of Several Coupled Positive-Definite Matrices

@article{Bertola2014UniversalityCA,
  title={Universality Conjecture and Results for a Model of Several Coupled Positive-Definite Matrices},
  author={Marco Bertola and Thomas Bothner},
  journal={Communications in Mathematical Physics},
  year={2014},
  volume={337},
  pages={1077-1141}
}
The paper contains two main parts: in the first part, we analyze the general case of $${p \geq 2}$$p≥2 matrices coupled in a chain subject to Cauchy interaction. Similarly to the Itzykson-Zuber interaction model, the eigenvalues of the Cauchy chain form a multi level determinantal point process. We first compute all correlations functions in terms of Cauchy biorthogonal polynomials and locate them as specific entries of a $${(p+1) \times (p+1)}$$(p+1)×(p+1) matrix valued solution of a Riemann… 

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References

SHOWING 1-10 OF 35 REFERENCES

Cauchy–Laguerre Two-Matrix Model and the Meijer-G Random Point Field

We apply the general theory of Cauchy biorthogonal polynomials developed in Bertola et al. (Commun Math Phys 287(3):983–1014, 2009) and Bertola et al. (J Approx Th 162(4):832–867, 2010) to the case

The Cauchy Two-Matrix Model

We introduce a new class of two(multi)-matrix models of positive Hermitian matrices coupled in a chain; the coupling is related to the Cauchy kernel and differs from the exponential coupling more

Universal microscopic correlation functions for products of independent Ginibre matrices

We consider the product of n complex non-Hermitian, independent random matrices, each of size N × N with independent identically distributed Gaussian entries (Ginibre matrices). The joint probability

Singular Values of Products of Ginibre Random Matrices

The squared singular values of the product of M complex Ginibre matrices form a biorthogonal ensemble, and thus their distribution is fully determined by a correlation kernel. The kernel permits a

Universal microscopic correlation functions for products of truncated unitary matrices

We investigate the spectral properties of the product of M complex non-Hermitian random matrices that are obtained by removing L rows and columns of larger unitary random matrices uniformly

A note on the limiting mean distribution of singular values for products of two Wishart random matrices

The product of M complex random Gaussian matrices of size N has recently been studied by Akemann, Kieburg, and Wei. They showed that, for fixed M and N, the joint probability distribution for the

Universal distribution of Lyapunov exponents for products of Ginibre matrices

Starting from exact analytical results on singular values and complex eigenvalues of products of independent Gaussian complex random N × N matrices, also called the Ginibre ensemble, we rederive the

Gap Probabilities for the Generalized Bessel Process: A Riemann-Hilbert Approach

We consider the gap probability for the Generalized Bessel process in the single-time and multi-time case, a determinantal process which arises as critical limiting kernel in the study of

Universality in the two‐matrix model: a Riemann‐Hilbert steepest‐descent analysis

The eigenvalue statistics of a pair (M1, M2) of n × n Hermitian matrices taken randomly with respect to the measure $${1 \over Z_{n}} \exp \left({-n} {\rm Tr}\,(V(M_{1})+ W(M_{2})- \tau

UNIFORM ASYMPTOTICS FOR POLYNOMIALS ORTHOGONAL WITH RESPECT TO VARYING EXPONENTIAL WEIGHTS AND APPLICATIONS TO UNIVERSALITY QUESTIONS IN RANDOM MATRIX THEORY

We consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn(x)dx = e−nV(x)dx on the line as n ∞. The potentials V are assumed to be real analytic, with