We present a set of conditions which, if satisfied, provide for a complete asymptotic analysis of random matrices with source term containing two distinct eigenvalues. These conditions are shown to be equivalent to the existence of a particular algebraic curve. For the case of a quartic external field, the curve in question is proven to exist, yielding precise asymptotic information about the limiting mean density of eigenvalues, as well as bulk and edge universality.
We present a supermatrix realisation of q-deformed spinor-spinor and spinor-vector R-matrices. These R-matrices are then used to construct transfer matrices for $U_{q^2}(\mathfrak{so}_{2n+1})$- and $U_{q}(\mathfrak{so}_{2n+2})$-symmetric closed spin chains. Their eigenvectors and eigenvalues are computed.
We propose a new approach to the spinor-spinor R-matrix with orthogonal and symplectic symmetry. Based on this approach and the fusion method we relate the spinor-vector and vector-vector monodromy matrices for quantum spin chains. We consider the explicit spinor R matrices of low rank orthogonal algebras and the corresponding RTT algebras. Coincidences with fundamental R matrices allow to relate the Algebraic Bethe Ansatz for spinor and vector monodromy matrices.
We prove the edge and bulk universality of random Hermitian matrices with equi-spaced external source. One feature of our method is that we use neither a Christoffel-Darboux type formula, nor a double-contour formula, which are standard methods to prove universality results for exactly solvable models. This matrix model is an example of a biorthogonal ensemble, which is a special kind of determinantal point process whose kernel generally does not have a Christoffel-Darboux type formula or double-contour representation. Our methods may showcase how to handle universality problems for biorthogon…
The Birkhoff polytope $\mathcal{B}_d$ consisting of all bistochastic matrices of order $d$ assists researchers from many areas, including combinatorics, statistical physics and quantum information. Its subset $\mathcal{U}_d$ of unistochastic matrices, determined by squared moduli of unitary matrices, is of a particular importance for quantum theory as classical dynamical systems described by unistochastic transition matrices can be quantised. In order to investigate the problem of unistochasticity we introduce the set $\mathcal{L}_d$ of bracelet matrices that forms a subset of $\mathcal{B}_d$,…
In the past we have considered Gaussian random matrix ensembles in the presence of an external matrix source. The reason was that it allowed, through an appropriate tuning of the eigenvalues of the source, to obtain results on non-trivial dual models, such as Kontsevich's Airy matrix models and generalizations. The techniques relied on explicit computations of the k-point functions for arbitrary N (the size of the matrices) and on an N-k duality. Numerous results on the intersection numbers of the moduli space of curves were obtained by this technique. In order to generalize these results to i…
A conjectural relationship between the GUE partition function with even couplings and certain special cubic Hodge integrals over the moduli spaces of stable algebraic curves is under consideration.
Random matrices with external source and multiple orthogonal polynomials
我们证明了随机厄米矩阵系综Z_n^{-1} \exp（-Tr（V（M）-AM））dM的平均特征多项式P_n（z）= E [ \det（zI-M）]由依赖于外部源A的特征值的多个正交性条件所表征.对于A的每个特征值a_j，都有一个权，P_n关于这个权有n_j个正交条件，如果n_j是a_j的重数，则特征值关联函数具有行列式形式，如Zinn-Justin所示.在这里，我们给内核一个不同的表达式。在A有两个不同的特征值的情况下，我们得到了一个Christoffel-Darboux公式，从而得到了一个由多个正交多项式满足的Riemann-Hilbert问题的紧致公式.
The Wishart model of random covariance or correlation matrices continues to find ever more applications as the wealth of data on complex systems of all types grows. The heavy tails often encountered prompt generalizations of the Wishart model, involving algebraic distributions instead of a Gaussian. The mathematical properties pose new challenges, particularly for the doubly correlated versions. Here we investigate such a doubly correlated algebraic model for real covariance or correlation matrices. We focus on the matrix moments and explicitly calculate the first and the second one, the compu…
We show that the Lipkin-Meshkov-Glick $2N$-fermion model is a particular case of one-spin Gaudin-type model in an external magnetic field corresponding to a limiting case of non-skew-symmetric elliptic $r$-matrix and to an external magnetic field directed along one axis. We propose an exactly-solvable generalization of the Lipkin-Meshkov-Glick fermion model based on the Gaudin-type model corresponding to the same $r$-matrix but arbitrary external magnetic field. This model coincides with the quantization of the classical Zhukovsky-Volterra gyrostat. We diagonalize the corresponding quantum Ham…
