A generalized plasma and interpolation between classical random matrix ensembles

Exact solution to the 1d one component Coulomb gas at fixed energy

The one dimensional one component plasma has applications to one dimensional particle systems with logarithmic interactions such as charges in a single channel wire or vortex filaments in a fluid convection stream. The exact integral of this plasma in the canonical ensemble with a gaussian confining potential has already been computed. In this paper, I compute the exact volume of the phase space of the plasma of N particles at fixed energy without a confining potential using a microcanonical ensemble and show that, as in the two-dimensional case, it has negative temperature states, suggesting …

A Solvable Two-Charge Ensemble on the Circle

We introduce an ensemble consisting of logarithmically repelling charge one and charge two particles on the unit circle constrained so that the total charge of all particles equals $N$, but the proportion of each species of particle is allowed to vary according to a fugacity parameter. We identify the proper scaling of the fugacity with $N$ so that the proportion of each particle stays positive in the $N \rightarrow \infty$ limit. This ensemble forms a Pfaffian point process on the unit circle, and we derive the scaling limits of the matrix kernel(s) as a function of the interpolating paramete…

Mixing of orthogonal and skew-orthogonal polynomials and its relation to Wilson RMT

The unitary Wilson random matrix theory is an interpolation between the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble. This new way of interpolation is also reflected in the orthogonal polynomials corresponding to such a random matrix ensemble. Although the chiral Gaussian unitary ensemble as well as the Gaussian unitary ensemble are associated to the Dyson index $β=2$ the intermediate ensembles exhibit a mixing of orthogonal polynomials and skew-orthogonal polynomials. We consider the Hermitian as well as the non-Hermitian Wilson random matrix and derive the corresponding…

Repulsion in low temperature $β$-ensembles

We prove a result on separation of particles in a two-dimensional Coulomb plasma, which holds provided that the inverse temperature $β$ satisfies $β>1$. For large $β$, separation is obtained at the same scale as the conjectural Abrikosov lattice optimal separation.

Muttalib--Borodin plane partitions and the hard edge of random matrix ensembles

We study probabilistic and combinatorial aspects of natural volume-and-trace weighted plane partitions and their continuous analogues. We prove asymptotic limit laws for the largest parts of these ensembles in terms of new and known hard- and soft-edge distributions of random matrix theory. As a corollary we obtain an asymptotic transition between Gumbel and Tracy--Widom GUE fluctuations for the largest part of such plane partitions, with the continuous Bessel kernel providing the interpolation. We interpret our results in terms of two natural models of directed last passage percolation (LPP):…

Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles

We compare finite rank perturbations of the following three ensembles of complex rectangular random matrices: First, a generalised Wishart ensemble with one random and two fixed correlation matrices introduced by Borodin and Péché, second, the product of two independent random matrices where one has correlated entries, and third, the case when the two random matrices become also coupled through a fixed matrix. The singular value statistics of all three ensembles is shown to be determinantal and we derive double contour integral representations for their respective kernels. Three different kern…

Asymptotic results in solvable two-charge models

Rider, Sinclair and Xu (2013) introduced a solvable two charge ensemble of interacting charged particles on the real line in the presence of the harmonic oscillator potential. It can be seen as a special form of a grand canonical ensemble with the total charge being fixed and unit charge particles being random. Moreover, it serves as an interpolation between the Gaussian orthogonal and the Gaussian symplectic ensembles and maintains the Pfaffian structure of the eigenvalues. A similar solvable ensemble of charged particles on the unit circle was studied by Shum and Sinclair (2014) and Forreste…

正交群和辛群中的布朗运动

Brownian Motion in Orthogonal and Symplectic Groups

开发统一图表框架刻画正交和辛随机矩阵的布朗系综，计算至多四阶多项式平均并构建正交不变插值，考虑其在量子信息、混沌及输运领域的应用。

Classical and Quantum Ensembles via Multiresolution. I. BBGKY Hierarchy

A fast and efficient numerical-analytical approach is proposed for modeling complex behaviour in the BBGKY hierarchy of kinetic equations. We construct the multiscale representation for hierarchy of reduced distribution functions in the variational approach and multiresolution decomposition in polynomial tensor algebras of high-localized states. Numerical modeling shows the creation of various internal structures from localized modes, which are related to localized or chaotic type of behaviour and the corresponding patterns (waveletons) formation. The localized pattern is a model for energy co…